HOW  TO  TEACH 

ARITHMETIC 


A  MANUAL  FOR  TEACHERS 

AND  A 

TEXT- BOOK  FOR  NORMAL  SCHOOLS 


BY 

JOSEPH  C.  BROWN 

HEAD  OF  DEPARTMENT  OF  MATHEMATICS,  HORACE  MANN 
SCHOOL,  COLUMBIA  UNIVERSITY 

AND 
LOTUS  D.  COFFMAN 

DEAN   OF  COLLEGE   OF   EDUCATION,   UNIVERSITY    OF    MINNESOTA 


CHICAGO  -  NEW  YORK 

ROW,  PETERSON  AND  COMPANY 


COPYRIGHT,  1914 
JOSEPH  C.  BROWN  AND  LOTUS  D.  COFFMAN 


CONTENTS 

PAET  ONE 

CHAPTER  PAGE 

I.     THE  HISTORY  OF  ARITHMETIC 1 

II.     SCIENTIFIC  STUDIES  IN  ARITHMETIC 17 

PAET  TWO 

III.  ACCURACY 43 

IV.  CHECKS  56 

V.     MARKING  PAPERS  IN  ARITHMETIC 66 

VI.  THE  NATURE  OF  PROBLEMS 71 

VII.  EULES  AND  ANALYSES 82 

VIII.  THE  VALUE  OF  DRILL 92 

IX.  WASTE  IN  ARITHMETIC 110 

PAET  THEEE 

X.     PRIMARY  ARITHMETIC 131 

XI.     THE  TEACHING  OF  THE  FUNDAMENTALS 148 

XII.     DENOMINATE  NUMBERS 171 

XIII.  COMMON  FRACTIONS 182 

XIV.  DECIMAL  FRACTIONS 195 

XV.     PERCENTAGE   213 

XVI.     THE  APPLICATIONS  OF  PERCENTAGE 223 

(a)  Profit  and  Loss. 

(b)  Commercial  Discount. 

(c)  Commission. 

(d)  Simple  Interest.     Annual  Interest. 
Compound  Interest. 

(e)  Insurance. 

(f)  Taxes  and  Eevenues. 

XVII.     BANKING,     CORPORATIONS,     STOCKS     AND     BONDS, 

BUSINESS  PRACTICE 271 

XVIII.     THE  METRIC  SYSTEM 282 

XIX.     INVOLUTION  AND  EVOLUTION 290 

XX.     EATIO  AND  PROPORTION 299 


374186 


VI 


CONTENTS 


CHAPTER  PAGE 

XXI.     MENSURATION 301 

XXII.     GRAPHS 319 

XXIII.  SHORT  CUTS 324 

XXIV.  LONGITUDE  AND  TIME 333 

XXV.     LITERAL  ARITHMETIC  AND  ALGEBRA 345 

XXVI.     PRESENT  TENDENCIES  IN  ARITHMETIC 349 

XXVII.     COURSES  OF  STUDY 367 

XXVIII.     BIBLIOGRAPHY  TOR  TEACHERS  OF  ARITHMETIC 368 

INDEX ...  371 


PREFACE 

This  book  was  written  for  the  purpose  of  improving 
the  teaching  of  arithmetic.  That  arithmetic  is  poorly 
taught  is  indicated  by  the  fact  that  a  larger  percentage 
of  children  fail  in  it  than  in  any  other  subject.  The  experi- 
ence of  the  authors  in  training  prospective  teachers  and  in 
institute  work  confirms  them  in  the  opinion  that  the  sub- 
ject is  suffering  partly  because  many  teachers  lack  instruc- 
tion in  its  theories,  methods,  and  devices.  The  authors 
do  not  assume  that  method  can  be  substituted  for  scholar- 
ship, but  they  do  contend  that  teachers  as  a  class  want  and 
need  definite  advice  in  the  teaching  of  arithmetic.  The 
demands  of  reading  circles,  of  special  methods  classes  in 
normal  schools  and  in  normal  high  schools,  and  of  super- 
visors, but  more  especially  the  needs  of  the  classroom 
teacher,  were  kept  in  mind  in  the  preparation  of  this  book. 

The  number  of  pages  given  to  the  discussion  of  a  topic 
is  no  index  of  the  relative  importance  of  the  topic  in  the 
curriculum;  it  rather  indicates  whether  a  subject  is  well 
or  poorly  taught.  However,  some  chapters  are  elaborated 
at  greater  length  than  others  for  the  reason  that  their 
material  is  intrinsically  or  historically  more  interesting. 

This  book  was  not  written  to  exploit  any  particular  text- 
book or  theory  or  method.  It  is  a  simple  and  accurate  expo- 
sition of  the  best  methods  employed  in  teaching*  the  sub- 
ject today. 

The  book  consists  of  three  parts :  Part  I  treats  the  his- 
tory of  arithmetic  and  the  contributions  recent  scientific 
studies  have  made  towards  standardizing  the  subject;  in 


PEEFACE 

Part  II  there  appears  a  treatment  of  certain  fundamental 
principles  and  ideas  that  apply  to  arithmetic  in  general; 
and  in  Part  III  the  methods  involved  in  teaching  the  vari- 
ous topics  or  divisions  of  the  ordinary  text-book  in  arith- 
metic are  described  in  detail. 

The  authors  gratefully  acknowledge  their  indebtedness 
to  Professor  David  Eugene  Smith,  of  Columbia  University ; 
Mr.  L.  C.  Lord,  President  of  the  Eastern  Illinois  Normal 
School;  Mr.  E.  Fiske  Allen,  of  the  State  Normal  School  at 
Emporia,  Kansas;  Mr.  G.  P.  Eandall,  Superintendent  of 
Schools,  Danville,  Illinois;  and  Miss  Lillian  Eogers,  of  the 
Horace  Mann  Elementary  School,  New  York  City,  for  read- 
ing certain  chapters  and  for  making  valuable  suggestions. 

J.  C.  B. 

L.  D.  C. 


HOW  TO  TEACH  ARITHMETIC 

PART  ONE 

CHAPTER  I 
THE  HISTORY  OF  ARITHMETIC 

Value  of  a  Knowledge  of  the  History  of  Arithmetic 

Chapters  and  books  showing  the  historical  development 
of  a  subject  of  study  are  frequently  given  scant  notice  and 
indifferent  attention,  especially  by  young  teachers.  Such 
an  attitude,  however,  never  leads  to  a  scholarly  knowledge 
of  the  subject.  With  the  demands  becoming  increasingly 
insistent  that  teachers  know  the  material  they  teach  sev- 
eral years  in  advance  of  their  students,  information  beyond 
the  lids  of  the  text-book  in  use  is  regarded  as  positive 
evidence  of  a  teacher's  professional  interest.  It  is  true  that 
a  wide  margin  of  scholarship  can  be  secured  by  studying 
the  more  advanced  phases  of  pure  mathematics.  It  is 
also  true  that  a  familiarity  with  the  historical  evolution  of 
any  subject  gives  a  perspective  that  enables  one  to  avoid 
errors,  reveals  methods  and  modes  of  practice  that  have 
proved  useful  or  futile,  and  teaches  direct,  economical  ways 
of  attacking  the  problems  of  the  subject.  In  arithmetic  we 
should  learn  the  folly  of  accepting  hasty  and  unwarranted 
conclusions  and  should  acquire  a  method  of  attacking 
directly  and  logically  a  set  of  conditions. 

In  these  days  of  educational  agitation,  the  newer  sub- 
jects are  regarded  by  many  as  having  the  greater  educa- 

1 


2  HOW  TO  TEACH  AEITHMETIG 

tional  value.  The  measure  of  the  educational  value  of  a 
subject  of  study  is  not  found  in  its  recency  of  origin.  That 
subject  which  has  persisted  throughout  a  comparatively 
long  period  of  time  and  which  is  essential  to  the  ordinary 
life  of  people  over  the  widest  extent  of  space  is  the  most 
fundamental.  Such  studies  are  not  subject  to  rapid 
change ;  their  growth  is  slow.  During  their  evolution  they 
have  ministered  to  the  satisfaction  of  the  common  needs  of 
the  common  people.  It  is  for  this  reason  that  they  are 
called  the  common  branches.  Some  branches  are  more  com- 
mon or  fundamental  than  others,  for  they  show  a  greater 
persistence  in  time  and  in  space.  No  wild  appeal  of  any 
educational  reformer  can  make  the  average  citizen — no  mat- 
ter whether  he  lives  in  San  Francisco,  New  Orleans,  New 
York,  London,  Paris,  Berlin,  or  St.  Petersburg, — believe 
that  knowledge  and  skill  in  arithmetic  are  not  necessary 
to  his  ordinary  life.  This  feeling  is  deep-seated  and 
ineradicable. 

Not  every  arithmetical  experience  has  been  saved  for 
the  present  generation  to  practice  upon.  Those  that  were 
trivial  and  insignificant,  or  were  of  use  only  to  special 
groups,  were  soon  discarded.  Most  of  those  that  were  saved 
were  the  important  experiences.  Their  importance  rested 
upon  the  degree  of  their  serviceableness  to  the  common 
people.  Those  found  in  the  text-books  of  to-day  represent 
the  important  experiences,  the  great  typical  experiences 
that  have  been  preserved  out  of  a  multitude  of  attempts  at 
environmental  adjustments. 

The  Slow  Growth  of  Arithmetic 

Arithmetic  was  not  born  in  a  day;  it  is  not  the  child  of 
some  fertile  mind;  it  is  not  the  creation  of  any  man,  nor 
of  any  given  group  of  men,  nor  of  any  sect  or  class  in 


THE  HISTORY  OF  ARITHMETIC  3 

society.  It  did  not  spring  fully  organized  into  being  be- 
cause groups  of  educational  seers  decided  on  some  bright 
day  that  the  children  of  the  earth  needed  a  new  study, 
and  they  would  organize  it  in  the  light  of  certain  a  priori 
principles.  On  the  contrary,  it  had  a  long  period  of 
evolution.  It  grew  by  piecemeal.  Its  beginnings  are 
enshrouded  in  the  mysteries  of  unwritten  history.  Its 
genesis  can  be  traced  to-day  in  some  of  the  practices  of 
primitive  man.  Of  this  we  are  reasonably  sure:  it  was 
born  out  of  necessity  and  its  more  or  less  disorganized 
fragments  were  used  for  strictly  utilitarian  ends.  It  was 
not  until  relatively  late  in  the  history  of  civilization  that 
arithmetic  became  one  of  the  philosopher's  chief  tools; 
and  it  was  not  until  the  Middle  Ages  that  it  became  a  hand- 
maiden of  theology.  Under  the  influence  of  the  philoso- 
phers and  the  monks  arithmetic  was  divorced  from  its 
utilitarian  sanctions  and  became  almost  wholly  a  subject 
for  pure  speculation. 

Why  Arithmetic  Has  Been  Taught 

Such  history  as  is  available  seems  to  warrant  the  con- 
clusion that  most  people  studied  arithmetic  and  sought  its 
extension,  not  for  any  peculiar  metaphysical  or  mind-  train- 
ing value  that  it  might  have,  but  because  it  was  needed 
in  the  practical  affairs  of  life.  Certainly  one  of  the  impor- 
tant reasons  for  teaching  arithmetic  was  that  it  was  needed 
in  the  trade  life  of  the  common  people.  For  this  reason  it 
was  frequently  urged  by  men  prominent  in  Grecian  and 
Roman  life  that  the  common  people  should  be  instructed 
in  only  the  simple  elements  of  calculation. 

Another  claim  for  arithmetic  was  that  it  sharpened  the 
intellect.  Curious  problems,  replete  with  catch  phrases, 
were  ingeniously  constructed  by  the  monks  as  material  for 


4  HOW  TO  TEACH  AEITHMETIC 

disputations.  One  needs  only  to  glance  through  educational 
journals,  periodicals,  and  mathematical  magazines  to  find 
evidence  of  the  persistence  of  this  mediaeval  tendency.  It 
was  presumed  that  arithmetic  had  peculiar  value  in  train- 
ing one  to  think  clearly,  quickly,  and  systematically.  But 
this  value  was  to  be  secured  by  making  the  subject  diffi- 
cult, by  making  it  obscure  and  unnecessarily  analytical. 

The  third  reason  for  giving  instruction  in  arithmetic 
was  the  conventional  and  cultural  value  of  the  subject. 
Solon,  Plato,  Aristotle,  and  Pythagoras  saw  that  arithmetic 
was  more  than  mere  calculation.  Pythagoras  believed  that 
only  through  the  study  of  arithmetic  could  one  become  per- 
fect and  fit  for  the  society  of  the  gods.  The  Humanists  of 
the  Middle  Ages,  although  they  added  nothing  to  the  sub- 
ject, upheld  the  claim  that  arithmetic  was  worthy  of  mas- 
tery for  its  own  sake;  i.  e.,  irrespective  of  any  use  to 
which  it  might  be  put. 

The  significant  fact  remains,  however,  in  the  face  of  all 
cultural,  conventional,  sentimental,  and  disciplinary  values 
that  may  be  set  upon  it,  that  arithmetic  has  flourished 
among  trading  peoples  and  has  found  its  real  justifica- 
tion there.  Wherever  a  new  gateway  to  trade  has  been 
opened  up,  wherever  a  new  center  of  commerce  has  been 
established,  there  arithmetic,  has  taken  on  new  life.  It  is 
essentially  a  commercial  subject.  It  has  accompanied  the 
march  of  civilization.  When  commercial  life  swept  from 
the  Orient  into  Europe,  arithmetic  accompanied  it;  when 
Italy  became  the  chief  trading  country  of  Europe,  arith- 
metic had  a  commercial  sanction  there;  when  trading  life 
became  common  among  the  Teutons  of  the  north,  arith- 
metic followed.  Practical  arithmetic  never  made  progress 
in  the  hands  of  the  scholastics  or  the  philosophers. 

Arithmetic,  like  every  other  subject  of  study,  has  evolved 
slowly.  Many  of  the  details  of  its  origin  are  hidden  in 


THE  HISTORY  OF  ARITHMETIC  5 

unravelled  history.  We  know  that  during  the  days  of 
antiquity  the  Babylonians,  the  Hindus,  and  the  Egyptians, 
made  rather  an  elaborate  use  of  number  forms  and  rela- 
tions. The  Babylonians  had  a  knowledge  of  arithmetical 
and  geometrical  progression  and  perhaps  of  proportion.  It 
is  quite  probable  that  they  used  the  abacus.  The  sex- 
agesimal system  which  they  used  may  have  been  derived 
from  the  division  of  the  year  into  360  days.  The  oldest 
treatise  on  mathematics  that  has  been  deciphered  was  writ- 
ten by  Ahmes,  an  Egyptian,  about  1700  B.  C.,  and  was 
based  upon  one  believed  to  date  as  far  back  as  3400  B.  C. 
This  manuscript  contains  a  number  of  arithmetical 
problems. 

Arithmetic  Among  the  Greeks  • 

The  Greeks  divided  the  science  of  arithmetic  into  two 
parts,  one  of  which  was  called  arithmetica  and  the  other 
logistica.  Arithmetica  treated  the  theory  of  numbers,  which 
were  believed  to  have  certain  mystical  values  and  were 
studied  and  classified  as  amicable,  deficient,  perfect,  and 
redundant.  Logistica  was  the  art  of  calculating;  mechan- 
ical devices  such  as  the  fingers,  counters,  and  the  abacus 
were  commonly  and  extensively  used  in  performing  its 
operations.  These  devices  later  disappeared  with  the  ap- 
pearance and  introduction  of  Arabic  or  Hindu  numerals. 
Dr.  Smith  thinks  that,  with  the  disappearance  of  these 
counting  devices,  there  was  a  corresponding  loss  of  power 
of  real  insight  into  number,  and  that  this  loss  remained 
almost  wholly  unrecognized  until  Pestalozzi  re-introduced 
object  teaching.1  Naturally,  the  Sophists  gave  much  atten- 
tion to  logistica,  while  Plato  considered  it  a  vulgar  and 
childish  art.  Arithmetica  was  the  tool  of  the  philosopher 

i  Smith,  "The  Teaching  of  Arithmetic, "  p.  58. 


6  HOW  TO  TEACH  ARITHMETIC 

»t 

while  logistica  was  the  tool  of  trade  life.  Consequently  the 
philosophers  looked  somewhat  with  disdain  upon  the  ser- 
vile tool  of  tradesmen. 

Arithmetic  Among  the  Romans 

The  Romans  contributed  practically  nothing  to  arith- 
metic, unless  it  be  their  cumbersome  system  of  notation 
which  has  already  passed  into  disuse.  Mathematicians 
speak  of  Roman  life  as  the  period  of  mathematical  sterility. 
No  European  country  had  a  better  opportunity  for  stand- 
ardizing arithmetic  and  giving  certain  methods  universal 
validity  than  the  Romans.  Although  they  had  great  prac- 
tical ability  for  organizing  armies  and  colonial  depend- 
encies, they  seemed  to  be  hopelessly  wanting  in  those  imagi- 
native and  speculative  powers  so  essential  to  true  scholar- 
ship. Moreover,  the  Romans  made  almost  no  attempt  to 
change  the  language,  customs,  and  traditions  of  their  colo- 
nial possessions.  The  quadrivium  consisted  of  arithmetic, 
music,  geometry,  and  astronomy.  Arithmetic  was  kept  alive 
through  'the  efforts  of  such  scholars  as  Boethius,  Isidore, 
St.  Boniface,  and  Alcuin.  The  ecclesiastics  used  it  to 
compute  Easter  and  other  movable  feasts,  and  for  training 
in  disputation.  The  problems  devised  for  this  latter  pur- 
pose represent  a  superlative  display  of  mathematical  in- 
genuity. 

The  Hanseatic  League 

The  Hanseatic  League,  although  it  was  organized  in  the 
thirteenth  century  to  protect  trade  routes  of  interest  to 
certain  cities,  soon  took  other  functions,  among  which  was 
the  general  improvement  of  commerce.  To  further  its 
purposes  it  established  Rechen  Schulen.  The  Rechenmeis- 
ters  organized  a  Guild  of  Rechenmeisters  which  was  largely 


THE  HISTOKY  OF  ARITHMETIC  7 

instrumental  in  keeping  arithmetic  out  of  the  schools  until 
it  was  introduced  by  Pestalozzi. 

Tine  Renaissance 

The  Renaissance  of  the  fifteenth  and  sixteenth  centuries 
received  a  rich  inheritance  in  arithmetic.  During  this  time 
the  contributions  of  earlier  nations  to  the  various  fields  of 
learning  were  eagerly  sought  after,  and  received  a  hitherto 
unknown  recognition.  Life  along  every  line  was  receiving 
a  new  impetus,  commerce  not  excepted.  Shipping  and  ship- 
building became  powerful  and  inviting  trades.  They  called 
forth  the  organization  of  great  stock  companies.  Business 
life  likewise  became  collectivistic.  Equation  of  payments 
came  into  general  use  and  many  elaborate  problems  in 
partnership  were  constructed.  Here  again  there  is  conclu- 
sive evidence  of  the  tendency  of  arithmetic  to  conform  to 
the  shifting  conditions  of  trade  life.  "The  sixteenth  cen- 
tury gave  us  printed  arithmetic,  with  important  transi- 
tions :  The  transition  from  the  use  of  the  Roman  symbols 
and  methods  to  the  use  of  the  Hindu  symbols  and  methods ; 
from  arithmetic  expressed  in  Latin  to  arithmetic  expressed 
in  the  language  of  the  reader;  from  arithmetic  in  manu- 
script to  arithmetic  in  the  printed  book;  from  arithmetic 
for  the  learned  to  arithmetic  for  the  people;  from  arith- 
metic theoretic  to  arithmetic  practical;  and  from  the  use 
of  counters  to  the  use  of  figures."  * 

Arithmetic  Since  the  Renaissance 

Advancement  since  the  Renaissance  has  been  even  more 
rapid.  Professor  Smith  considers  progress  to  have  been 
made  mainly  along  six  lines:  (1)  a  marked  evolution  of 
the  commercial  phases  of  arithmetic,  (2)  the  invention  of 

i  Jackson  "The  Educational  Significance  of  Sixteenth  Century 
Arithmetic, ' '  Teachers'  College,  Columbia  University,  p.  24. 


8      -  HOW  TO  TEACH  AEITHMETIC 

common  symbols  of  operation  probably  between  1550  and 
1650,  (3)  the  invention  of  decimal  fractions  about  1600, 
(4)  the  invention  of  logarithms  by  Napier  in  1614,  (5) 
the  modification  and  improvement  of  methods  of  multi- 
plying and  dividing  and  the  use  of  the  Austrian  method  of 
subtracting  and  multiplying,  (6)  the  introduction  and  im- 
provement of  algebraic  symbols,  with  the  introduction  of 
which  certain  subjects  such  as  alligation,  series,  rule  of 
three,  and  roots  either  disappeared  or  began  to  disappear 
from  ;the  arithmetics,  because  of  the  greater  ease  with 
which  they  could  be  solved  by  algebra.1 

Racial  Development  of  Fundamental  Phases  of  Arithmetic 

Counting  was  the  initial  step  in  the  development  of  arith- 
metic by  the  race.  The  earliest  possible  glimmering  of  the 
race's  intelligence  of  arithmetic  must  have  come  with  the 
counting  of  like  things.  This  one  to  one  correspondence  is 
the  fundamental  concept  of  arithmetic.  Computation  con- 
sisted of  nothing  more  than  the  simple  counting  of  like 
things  until  the  race  recognized  that  one  object  might  be 
considered  the  equivalent  in  value  of  several.  Then  and 
not  until  then  did  arithmetic  really  begin.  Some  light  is 
shed  upon  the  rudimentary  character  of  the  number  sense 
by  a  study  of  the  attainments  of  primitive  peoples  to-day. 
The  almost  universal  tool  used  by  them  for  counting  is  the 
fingers.  Whenever  sticks,  splints,  pebbles,  shells,  notches 
in  sticks,  or  knots  in  strings  are  used  as  practical  methods 
of  numeration,  their  common  origin  is  explained  by  the 
universal  finger  method.  The  use  of  the  fingers  probably 
accounts  for  10  being  the  basis  of  our  number  system. 

Just  as  the  race  communicated  by  oral  language  before 
it  wrote,  so  it  counted  before  it  invented  a  system  of  nota- 

"* Smith,  "The  Teaching  of  Elementary  Mathematics, "  ppr.  66-68. 


THE  HISTORY  OF  ARITHMETIC  9 

tion.  One  of  the  most  interesting  chapters  in  the  history 
of  arithmetic  is  that  which  treats  the  various  systems  of 
number  symbolism.  A  great  number  of  systems  were  in- 
vented. One  can  almost  trace  the  intellectual  development 
of  the  race  in  the  growth  of  systems  of  notation.  It  is  well 
known  that  primitive  man  proceeded  beyond  5  in  counting 
with  the  very  greatest  difficulty.  Beyond  that  he  used 
such  terms  as  much,  many,  a  heap,  or  plenty,  instead  of 
definite  terms  referring  to  quantities  in  a  series. 

The  evolution  and  extension  of  a  series  of  numbers  into 
a  system  was  not  possible  until  some  number  had  been 
agreed  upon  and  accepted  as  a  base.  Only  here  and  there 
did  the  savage  mind  grasp  this  fundamental  notion.  It 
was  his  customary  practice  to  illustrate  each  successive  step 
with  a  particular  object.  If  it  could  not  be  illustrated 
satisfactorily,  he  simply  referred  to  it  as  many  or  a  heap. 
So  long  as  his  counting  did  not  exceed  the  number  of  fin- 
gers on  his  two  hands  a  system  was  unnecessary.  But  when 
it  became  necessary  to  think  ten  and  one,  and  to  supply 
some  independent  term  for  the  idea,  ten  became  the  basis 
of  his  number  system.  Of  course,  in  the  case  of  the 
quinary  system  the  base  was  established  at  5,  and  in  the 
vigesimal  system  at  20.  Other  numbers  have  served  as 
points  of  departure  for  the  establishment  of  systems,  for 
example  as  in  the  binary  system  of  Liebnitz,  the  ternary 
and  quaternary  systems  of  the  Haida  Indians  of  British 
Columbia.  There  is  probably  no  recorded  instance  of  a 
number  system  formed  on  6,  7,  8,  or  9,  as  a  base.1 

The  Writing  of  Numbers 

The  symbolic  representation  and  meaning  of  the  terms 
in  the  different  systems  varied  greatly  among  different  na- 

iConant,    "The    Number    Concept, "  Macmillan,  1896,  p.  119. 


10  HOW  TO  TEACH  ARITHMETIC 

tions.  The  Egyptians  had  symbols  for  1,  10,  100,  and  the 
higher  powers  of  10;  the  Babylonians  had  symbols  for  1, 
10,  and  100;  the  Greeks  indicated  the  numbers  by  giving 
the  initial  letters  of  their  names ;  the  Romans  resorted  to 
the  letters  of  an  old  Greek  alphabet.  All  of  these  plans 
failed  to  survive,  because  they  did  not  contain  a  symbol 
to  represent  zero  or  naught.  This  greatest  of  all  mathe- 
matical inventions  came  from  the  Hindus.  It  is  sometimes 
erroneously  ascribed  to  the  Arabians,  but  recent  investiga- 
tions show  rather  conclusively  that  the  Arabs  borrowed  this 
idea  as  well  as  most  of  their  notary  scheme  from  the 
Hindus.  The  invention  of  zero  and  its  application  gave 
place  value  to  the  number  series,  a  fundamental  fact  that 
was  largely  absent  from  all  previous  schemes. 

Common  Fractions 

Perhaps  the  third  great  step  in  the  evolution  of  arith- 
metic following  those  of  counting  and  notation,  was  the 
invention  of  simple  fractions.  There  is  convincing  evidence 
that  they  were  quite  generally  used  at  an  early  date.  It 
seems  logical  to  infer  that  number  ideas,  and  a  number 
language  of  integral  quantities  and  relationships,  must 
have  preceded  in  time  a  system  of  fractional  enumeration. 
Centuries  were  required  before  any  adequate  or  satisfac- 
tory system  was  evolved.  The  cumbersome  character  of 
the  early  methods  is  described  more  at  length  in  the  chap- 
ter on  common  fractions.  The  Babylonians  kept  the  de- 
nominator (60)  constant  and  changed  the  numerator;  the 
Romans  kept  the  denominator  (12)  constant,  and  changed 
the  numerator;  the  Egyptians  and  the  Greeks  kept  the 
numerator  constant  and  changed  the  denominator.  Com- 
mon fractions  were  in  vogue  centuries  before  decimal 
fractions  were  invented. 


THE  HISTORY  OF  ARITHMETIC  11 

Decimal  Fractions 

The  last  fundamental  tool  for  arithmetical  work  to  be 
invented  was  decimal  fractions.  They  did  not  appear  until 
the  sixteenth  century.  They  were  used  as  early  as  1610 
and  sporadically  thereafter  until  the  eighteenth  century, 
when  they  were  included  for  the  first  time  as  a  part  of  the 
regular  school  work,  although  they  were  not  currently  used 
until  the  nineteenth  century. 

Counting,  notation,  common  and  decimal  fractions,  make 
possible  all  arithmetical  operations.  Perhaps  the  order  in 
which  they  evolved  is  indicative  of  the  order  in  which  they 
should  be  taught.  At  any  rate,  it  is  indicative  of  the  rela- 
tion which  they  bear  to  .one  another. 

Method  in  Arithmetic 

Instruction  in  arithmetic  was  based  upon  object  teach- 
ing until  the  sixteenth  century  when  the  Hindu-Arabic 
numerals  were  introduced.  The  introduction  of  these 
numerals  was  followed  by  a  marked  improvement  in  cal- 
culation, and  a  consequent  neglect  of  the  philosophy  of 
arithmetic.  Text-books  were  filled  with  a  multitude  of 
rules  and  examples.  Naturally,  the  work  became  extremely 
formal  and  mechanical.  It  would  be  interesting,  if  not 
valuable,  to  trace  the  persistence  of  traditions  for  routine 
work  established  then  in  the  changing  character  of  text- 
books and  methods  down  to  the  present  time.  It  is  certain 
that  arithmetic  is  still  taught  in  some  schools  by  the  com- 
mitting of  rules  and  the  mere  solving  of  problems. 

Eventually  a  reaction  set  in  against  the  deadening  uni- 
formity in  whose  clutches  method  was  held.  One  of  the 
earliest  attempts  to  break  it  up  and  to  reduce  the  drudgery 
involved  in  learning  rules,  was  the  publication  of  rhyming 


12  HOW  TO  TEACH  AEITHMETIC 

arithmetics.    These  books  became  very  common  during  the 
seventeenth  century. 

During  the  long,  dark  period  of  teaching  without  objects, 
the  characteristic  mode  of  instruction  was  individualistic. 
Pupils  received  personal  help  from  the  teacher  whenever 
it  was  necessary.  They  copied  their  own  problems  and 
solved  them  in  silence.  There  was  little  or  no  class  work. 
There  is  no  way  of  knowing  whether  such  personal,  indi- 
vidual instruction  called  forth  more  ability  and  produced 
a  higher  quality  of  attainment  than  that  produced  by  the 
present  modes  of  group  instruction,  but  that  it  had  its 
inherent  strength  and  advantages  is  evidenced  by  the  pres- 
ent tendency  to  reduce  the  size  of  classes. 

Von  Busse 

Gottlieb  von  Busse,  1786,  was  prpbably  the  first  to  use 
number  pictures  (Zahlenbilder)  systematically.  He  ar- 
ranged them  according  to  this  plan  : 


5 

five 

It  was  a  short  and  easy  step  from  the  size  of  the  num- 
ber pictures  to  a  liberal  use  of  objects. 

Pestalozzi 

Pestalozzi  was  the  first  to  recognize,  appreciate  and 
utilize  the  full  value  of  objects  in  arithmetic.  He  did 
not  approve  of  the  abacus  or  of  geometric  forms  in 
teaching  arithmetic.  The  re-introduction  of  objects  meant 
not  only  the  restoration  of  the  teaching  of  numbers  be^- 


THE  HISTORY  OF  ARITHMETIC  13 

fore  figures,  but  it  also  operated  to  free  children  from 
slavish  bondage  to  rules.  The  great  concern  and  reverence 
of  Pestalozzi  for  childhood  expressed  itself  in  a  definite 
attempt  to  adapt  instruction  to  the  mental  powers  of  chil- 
dren. To  him  the  only  way  to  the  pedagogical  heaven  was 
through  object  teaching.  This  resulted  in  a  new  and  in- 
creased emphasis  upon  oral  arithmetic.  The  subject  in- 
creased in  popularity  until  it  secured  the  most  prominent 
place  in  the  curriculum. 

Tillich 

Of  the  disciples  of  Pestalozzi  perhaps  Tillich  was  the 
most  noted.  He  continued  to  simplify  the  methods  and 
materials  of  his  great  master  (1)  by  placing  greater  em- 
phasis upon  the  relations  of  the  number  forms  1  to  10;  (2) 
by  showing  that  any  number  of  two  digits  might  be  con- 
sidered as  so  many  tens  and  so  many  units;  and  (3)  by 
inventing  a  Reckoning-chest,  which  contained  10  one-inch 
cubes,  10  parallelepipeds  two  inches  high  and  one  inch 
square,  10  three  inches  high,  and  so  on  to  those  ten  inches 
high.  By  means  of  these  blocks  he  taught  notation  and  the 
relation  of  tens  and  units.  Mental  arithmetic  was  made 
the  basis  of  his  work.  He  exercised  more  wisdom  than 
Pestalozzi  in  the  selection  of  suitable  materials  for  percep- 
tion, and  in  recognizing  the  decimal  idea. 

Frederick  Kranckes 

In  a  book  by  Frederick  Kranckes,  which  appeared  in 
1819,  there  is  advocated  for  the  first  time  the  teaching  of 
arithmetic  by  the  concentric  circle  plan.  He  recommended 
four  circles,  the  first  to  contain  the  number  relations  from 
1  to  10,  the  second  from  1  to  100,  the  third  from  1  to  1000, 
and  the  last  from  1  to  10000.  Like  Busse,  he  made  exten- 


14  HOW  TO  TEACH  ARITHMETIC 

sive  use  of  number  pictures.  His  book  was  of  a  more  dis- 
tinctly pedagogical  character  than  those  of  his  contem- 
poraries, for  he  planned  for  the  development  of  the  rules 
and  principles  of  arithmetic  through  intellectual  instruc- 
tion. In  this  latter  respect  he  improved  upon  the  teachings 
of  Pestalozzi,  whose  oral  work  was,  like  that  of  the  two 
preceding  centuries,  still  absurdly  formal  and  abstract. 

Grube 

Grube  (1810-1884)  seized  upon  Kranekes'  idea  of  con- 
centric circles,  but  reduced  the  scheme  to  two  circles ;  the 
first  of  which  included  a  year 's  work  on  the  relations  from 
1  to  10.  This  spiral  plan  became  the  basic  organizing  prin- 
ciple of  many  arithmetics  in  America. 

Although  Grube  used  objects  more  elaborately  than  any 
of  his  predecessors,  he  evolved  no  new  principle  governing 
their  application.  The  only  new  idea  he  advanced  was  that 
all  the  fundamental  operations  should  be  taught  in  connec- 
tion with  each  number  before  the  next  number  was  taken 
up.  In  the  seventeenth  century  each  topic  was  completed 
in  turn.  Now  we  have  the  other  extreme,  all  the  number 
processes  taught  together  before  the  next  number  is 
introduced. 

The  two  chief  criticisms  against  Grube  have  been  his  too 
liberal  use  of  objects  and  his  advocacy  of  the  simultaneous 
mastery  of  all  the  fundamental  processes.  His  point  of 
view  as  to  the  second  of  these  is  neither  natural  nor 
psychological.  It  is  not  natural  because  it  does  not  accord 
with  the  racial  evolution  of  these  processes  nor  with  the 
relative  difficulties  inherent  in  the  various  operations.  It 
is  not  psychological  because  it  is  a  distinct  attempt  to  re- 
verse the  pedagogical  maxim,  "We  must  proceed  from  the 
simple  to  the  complex/' 


TTJE  HISTORY  OF  ARITHMETIC  15 

Object  Teaching  Over-dome 

No  experienced  teacher  or  supervisor  questions  the  value 
of  object  teaching  in  the  primary  school.  It  is  not  only 
desirable  but  necessary  that  the  pupil  should  be  aided  in 
his  grasp  of  number  by  approaching  it  from  the  concrete. 
However,  any  plan  or  method  becomes  detrimental  to  the 
best  interests  of  the  pupil  if  it  be  over-emphasized.  We 
are  frequently  reminded  that  the  educational  pendulum 
oscillates  between  extremes.  This  has  been  true  in  object 
teaching  in  arithmetic.  Some  enthusiasts  have  carried 
object  teaching  to  absurd  extremes  and  as  a  result  the 
pupils  have  been  retarded  rather  than  aided  in  their  mas- 
tery of  number  relations. 

The  writers  are  familiar  with  a  city  system  where  the 
plan  of  teaching  all  number  ideas  by  measures  and  pictures 
was  carried  to  its  logical  extreme.  The  children  through 
the  fourth  grade  were  required  to  solve  all  problems  by" 
concrete  demonstration  or  pictorial  illustrations.  A  com- 
prehensive examination  of  these  same  pupils  was  made  when 
they  reached  the  sixth  grade,  and  by  comparing  the  results 
with  those  obtained  from  sixth  grade  children  in  other 
schools,  the  conclusion  was  unmistakable  that  the  children 
taught  by  this  method  had  lost  almost  a  full  year.  They 
had  been  arrested  upon  the  plane  of  concrete  thinking  in 
arithmetic.  They  had  not  learned  to  think  in  symbols,  nor 
had  they  acquired  an  automatic  mastery  of  the  fundamen- 
tal operations.  We  should  be  open  to  criticism  if  we  gen- 
eralized upon  a  single  case,  but  the  results  in  this  instance 
have  been  corroborated  by  superintendents  elsewhere. 
Instead  of  the  method  producing  better  mathematicians, 
the  pupils  taught  by  it  knew  less  arithmetic,  were  less 
facile  and  accurate  in  computation  than  other  children  of 
equal  age,  maturity,  and  training.  This  fundamental  defect 


16  HOW  TO  TEACH  ARITHMETIC 

in  the  claims  made  for  it,  we  believe,  accounts  for  its 
failure. 

The  Purpose  of  this  Book 

The  most  cautious  as  well  as  the  most  able  authors  have 
recently  been  attempting  to  organize  the  extreme  claims  of 
earlier  theorists  into  a  rational  scheme.  It  is  true,  the 
expressions,  "the  laboratory  method  and  correlation,"  are 
sometimes  used  in  discussions  on  the  teaching  of  arithmetic, 
but  these  terms  connote  methods  as  applicable  to  the  teach- 
ing of  other  subjects  as  to  arithmetic.  It  is  the  intention 
of  the  authors  of  this  book  to  gather  up  these  stray  threads 
of  method  and  to  relate  them  to  the  best  teaching  practice 
of  the  day. 


CHAPTER  II 

SCIENTIFIC  STUDIES  IN  ARITHMETIC 

There  is  a  tendency  in  education  for  experimental  study 
to  supplant  dogmatic  statement.  Hitherto  progress  has  been 
made  by  the  laborious  process  of  trial  and  error.  "What 
seemed  to  be  successful  experience  has  been  the  criterion  for 
judging  practice.  But  within  the  last  decade  a  restless  dis- 
content with  the  use  of  opinion  as  the  final  standard  for 
evaluating  methods  and  materials  has  invaded  every  field 
of  human  activity.  The  breakdown  of  old  authoritative 
forms  of  control  has  been  accompanied  by  an  increased 
interest  in  experiment,  investigation,  and  science.  The 
prevailing  tendency  to  manage  commerce,  business,  and 
manufacturing  scientifically,  has  expressed  itself  in  educa- 
tion in  the  attempt  to  establish  units  and  scales  for  meas- 
uring educational  products.  An  elaborate  statistical 
technique  is  now  employed  by  the  specialist  in  education 
in  making  his  investigations  and  in  preparing  his  scales. 
It  is  not  our  purpose  to  discuss  the  nature  of  this  technique 
nor  the  reliability  of  the  results  secured.  Comparatively 
few  of  these  studies  have  been  made  in  the  field  of  arith- 
metic, but  these  few  have  illuminated  the  whole  field. 

Summary  of  Rice's  Investigation 

The  first  attempt  to  measure  and  to  evaluate  the  teaching 
of  arithmetic  was  made  by  Mr.  J.  M.  Rice,1  in  which  he 
sought  answers  to  three  questions :  What  results  should  be 
accomplished  whenever  a  subject  is  incorporated  in  the 

i  Forum  34:281-297;  437-452. 

17 


18  HOW  TO  TEACH  ARITHMETIC 

school  program?  How  much  time  shall  be  devoted  to 
the  branch?  Why  do  some  schools  succeed  in  securing 
satisfactory  results  with  a  reasonable  appropriation  of  time, 
while  others  cannot  get  reasonable  results  with  a  satisfac- 
tory appropriation  of  time  ? 

Mr.  Rice's  test,  given  in  1902,  consisted  of  a  series  of 
eight  problems,  given  to  6000  pupils  in  the  fourth  to 
eighth  school  years  inclusive.  The  basis  used  by  Mr.  Rice 
in  selecting  these  problems  and  the  manner  in  which  he 
scored  the  manuscripts  were  not  explained  fully  enough  to 
enable  other  investigators  to  duplicate  the  study.  However, 
he  arrived  at  the  following  conclusions : 

1.  Variation  in  ability  as  shown  by  the  ratings  given 
individual  children  is  common  to  all  grades,  but  it  is  greater 
in  the  seventh  and  eighth  year  classes  than  in  the  earlier 
ones.    The  range  in  ability  in  the  seventh  year  is  from  8.9 
to  81.1%,  and  in  the  eighth  year  from  11.3  to  91.7%. 

2.  Superiority,  mediocrity  or  inferiority  in  any  grade  in 
a  given  city  in  general  means  superiority,  mediocrity  or 
inferiority  in  all  grades. 

3.  Differences  in  home  environment  do  nqt  explain  these 
differences  in  attainment.  Of  the  eighteen  schools  examined, 
three   in  particular  were   representative   of  the   "aristo- 
cratic" districts,  and  the  best  of  these  ranked  tenth,  while 
the  others  ranked  eleventh  and  sixteenth  respectively.    The 
school  that  ranked  seventh  was  located  in  a  slum  district. 
The  fifth  school  in  rank  was  in  a  district  that  is  but  a 
"shade  better  than  those  of  the  slums/' 

4.  Differences  in  attainment  are  not  explained  by  dif- 
ferences in  size  of  classes.    In  general  the  number  of  pupils 
per  class  was  large  in  the  schools  that  ranked  highest,  and 
small  in  the  schools  that  ranked  lowest. 

5.  Results  in  arithmetic  are  correlated  with  maturity; 


SCIENTIFIC  STUDIES  IN  ARITHMETIC  19 

that  is,  the  averages  improve  from  grade  to  grade.  That 
the  mere  fact  of  age  is  not  sufficient  to  account  for  dif- 
ferences in  results  is  shown  by  the  following  table : 

FOURTH  GRADE   FIFTH  GRADE    SIXTH  GRADE 
Average    Per     Average    Per     Average    Per 
age        cent        age        cent        age    *   cent 

Six  highest  schools 11.9       62.8       12.6       84.3       13.4       96.3 

Six  lowest  schools 11.0       29.0       12.0       49.8       13.4       61.4 

SIXTH  GRADE  SEVENTH  GRADE  EIGHTH  GRADE 

Average    Per     Average    Per     Average    Per 

age.       cent        age        cent        age        cent 

Five  highest  schools 13.4       49.5       14.1       71.9       14.1       90.4 

Five  lowest  schools. .        .   13.4       11.0       13.11     29.0       14.5       38.0 


6.  The  results  indicate  that  the  time  of  day  at  which  the 
tests  were  given  was  not  responsible  for  the  differences  in 
grades  received. 

7.  There  is  no  direct  relation  between  time  and  results. 
"The  amount  of  time  devoted  to  arithmetic  in  the  school 
that  obtained  the  lowest  average,  25  per  cent,  was  prac- 
tically   the    same    as    in    the    one    where    the    highest 
average,  80  per  cent,  was  obtained.     In  the  former  the 
regular  time  for  arithmetic  in  all  grades  was  forty-five 
minutes  a  day,  but  some  additional  time  was  given.     In 
the  latter  the  time  varied  in   the   different  classes,   but 
averaged  fifty-three  minutes  daily. 

"From  these  few  facts  two  important  deductions  may 
be  made:  First,  that  unsatisfactory  results  cannot  be 
accounted  for  on  the  ground  of  insufficient  instruction; 
and  second,  that  the  schools  showing  the  favorable  results 
cannot  be  accused  of  having  made  a  fetish  of  arithmetic. 
These  statements  are  further  justified  by  the  fact  that  the 
four  schools  which,  on  the  whole,  stood  highest,  ga*ve  prac- 


20  HOW  TO  TEACH  AEITHMETIC 

tically  the  same  amount  of  time  to  arithmetic  as  the  three 
schools  which  stood  lowest/' 

8.  The  amount  of  home  work  required  is  no  criterion  of 
the  results  to  be  expected  in  school. 

i  l  By  far  the  greatest  amount  of  home  work  in  arithmetic 
was  required  in  the  city  whose  schools  obtained  the  poorest 
results."  On  the  other  hand,  the  five  cities  standing  high- 
est had  practically  abandoned  home  work. 

9.  Methods  in  teaching  are  not  the  controlling  element 
in  the  accomplishment  of  results. 

In  all  schools  tested  by  Mr.  Rice  thoroughly  modern 
methods  were  used. 

10.  Variations  in  results  cannot  be  accounted  for  by 
differences  in  the  general  qualifications  of  teachers. 

The  results  in  a  given  city  were  nearly  all  good,  mediocre 
or  bad;  extreme  variations  did  not  appear  among  the  dif- 
ferent classrooms  of  the  same  locality,  although  the  teachers 
differed  greatly  in  the  amount  of  training  they  had 
received. 

11.  Mr.  Rice  reaches  the  final  conclusion  that  the  super- 
visor is  the  controlling  factor  determining  differences  in 
achievement  in  arithmetic.     Although  one  of  the  primary 
functions  of  a  supervisor  is  the  preparation  of  a  course  of 
study,  it  was  clearly  evident  that  the  excellence  of  the 
course  of  study  would  not  enable  one  to  prophesy  as  to  the 
achievements  of  the  pupils.     The   important   factors  by 
which  the  supervisor  accomplishes  these  results  are  the 
standards  and  tests  that  he  uses.     In  general,  the  imme- 
diate and  most  potent  device  used  by  supervisors  in  those 
schools  that  ranked  best  was  an  examination  to  test  the 
teacher's  progress.     These  examinations  emphasized  both 
the  reasoning  and  routine  sides  of  arithmetic.     They  were 
suggestive,  and  stimulated  teachers  to  secure  better  results. 


SCIENTIFIC  STUDIES  IN  ARITHMETIC  21 

"Summary  of  Stone's  Investigation" 

Dr.  C.  W.  Stone's  study  on  "Arithmetical  Abilities  and 
Some  of  the  Factors  Determining  Them"1  is  one  of 
the  most  significant  investigations  of  a  scientific  character. 
This  study  was  a  definite  attempt  to  find  the  nature  of  the 
product  of  the  first  six  years  of  arithmetic,  and  the  relation 
between  certain  distinctive  procedures  and  the  resulting 
abilities.  Dr.  Stone  personally  collected  data  from  twenty- 
six  representative  school  systems,  distributed  over  New 
England,  the  Middle  States,  and  the  Central  West.  Two 
series  of  tests  were  given  to  the  6A  grade  in  each  school 
system  examined,  one  in  fundamentals  and  one  in  rea- 
soning. 

In  these  tests  given  by  Dr.  Stone  a  time  limit  of  twelve 
minutes  was  set  for  the  solution  of  the  problems  in  funda- 
mentals, and  fifteen  minutes  for  those  in  reasoning.  The 
time  allotted  in  each  case  was  too  little  for  even  the  bright- 
est pupil  to  solve  all  of  the  problems.  This  plan  does  not 
correspond  with  current  practice  in  the  holding  of  exami- 
nations. The  customary  practice  is  to  give  a  few  prob- 
lems with  the  expectation  that  every  one  will  solve  them 
all.  This  gives  some  measure  of  the  proficiency  of  the 
class,  but  it  does  not  give  a  measure  of  the  individual 
abilities  of  the  pupils.  The  rating  of  individuals  as  to  their 
facility  in  handling  fundamentals  or  ability  to  solve  prob- 
lems involving  reasoning  is  best  shown  when  no  one  has 
time  to  solve  all  of  the  examples  or  problems  in  the  test. 
One  such  test  would  show  individual  differences  better  than 
no  test  at  all,  but  the  average  of  several  tests  would  give  a 

i  A  thorough  comprehension  of  this  study  can  be  secured  only  by 
an  examination  of  the  entire  tables.  Erroneous  impressions  will  thus 
be  prevented.  Published  by  Teachers'  College,  Columbia  University, 
New  York  City. 


22  HOW  TO  TEACH  ARITHMETIC 

much  more  accurate  rating  of  the  individuals  of  the  class. 
Examinations  of  the  kind  Dr.  Stone  used  will  come  into 
more  common  use  in  proportion  as  teachers  become  more 
skillful  in  Devaluating  methods  and  results. 

Tests  of  this  kind  lose  much  of  their  value  unless 
there  is  a  uniform  system  for  scoring  the  results.  The 
standards  used  to-day  vary  with  localities,  with  the  train- 
ing and  experience  of  the  teacher,  with  her  inclinations, 
temperamental  attitudes,  and  personal  knowledge  of  chil- 
dren. One  teacher  grades  on  method,  so  much  for  trying, 
another  on  the  correctness  of  the  result ;  one  takes  into  con- 
sideration neatness,  and  another,  punctuation.  Obviously 
there  are  many  factors  operating ;  to  speak  of  standards  is 
really  anomalous;  we  cannot  get  them  until  we  agree  to 
use  uniformly  certain  common  units.  No  matter  how  much 
we  may  question  Dr.  Stone's  method  of  checking  results, 
if  we  wish  to  compare  any  school  with  his  we  should 
use  the  same  method.  In  addition  a  score  of  one  was  given 
for  each  column  added  correctly,  and  in  multiplication  a 
score  of  one  was  given  for  each  correct  partial  product,  and 
for  each  column  added  correctly.  This  same  plan  was  pur- 
sued in  scoring  the  examples  in  subtraction  and  division. 
Two  methods  were  used  in  scoring  the  problems  in  reason- 
ing :  after  the  problems  had  been  arranged  in  the  order  of 
relative  difficulty,  a  score  was  given  for  each  problem  rea- 
soned correctly  on  the  basis  of  its  relative  difficulty  or  only 
the  first  six  problems,  which  all  children  had  time  to  solve, 
were  scored. 

The  problems  were  arranged  as  to  their  relative  diffi- 
culty by  giving  one  hundred  sixth  grade  pupils  time  enough 
to  solve  all  of  the  problems  they  could  in  the  order  printed. 

The  problems  were  next  given  in  reverse  order  to  one 
hundred  pupils  who  had  as  much  time  as  they  wished  to 
solve  them.  The  average  showed  the  order  of  difficulty. 


SCIENTIFIC  STUDIES  IN  ARITHMETIC  23 

What  the  Scores  Measure 

"As  used  in  this  study  the  achievements,  products/ and 
abilities,  except  where  otherwise  qualified,  must  necessarily 
refer  to  the  results  of  the  particular  tests  employed  in  this 
investigation.  That  some  systems  may  achieve  other  and 
possibly  quite  as  worth  while  results  from  their  arithmetic 
work  is  not  denied;  but  what  is  denied  is  that  any  system 
can  safely  fail  to  attain  good  results  in  the  work  covered 
by  these  particular  tests.  Whatever  else  the  arithmetic 
work  may  produce,  it  seems  safe  to  say  that  by  the  end  of 
the  sixth  school  year  it  should  result  in  a  good  ability  in 
the  four  fundamental  operations  and  the  simple  every-day 
kind  of  reasoning  called  for  in  these  problems.  It  does  not 
then  seem  unreasonable,  in  view  of  the  precautions  pre- 
viously enumerated,  to  claim  that  the  scores  made  in  the 
respective  systems  afford  a  reliable  measure  of  the  products 
of  their  respective  procedures  in  arithmetic/'  Stone,  p.  19, 

Relative  Difficulty  of  Column  Addition 

/ 

It  is  usually  assumed  that  addition  increases  in  diffi- 
culty in  proportion  as  the  columns  increase  in  length.  Dr. 
Stone  found  that  "one  step  in  a  problem  in  fundamentals 
is  about  equal  to  another,  be  the  step  long  or  short ;  e.  g., 
96  per  cent  of  the  children  did  the  first  column  of  problem 
one  correctly  "(six  numbers  to  the  column),  and  exactly 
the  same  per  cent  did  the  longer  and  presumably  harder 
column  of  problem  four  correctly  (eight  numbers  to  the 
column).  Similarly,  about  as  many  pupils  failed  in  the 
very  short  additions  of  the  partial  products  in  the  multi- 
plication problems  as  failed  in  the  long  columns  of  the 
multiplication  problems/"  (p.  16,  17)  The  difficulty  that 
children  are  supposed  to  have  with  columns  of  extra  length 
is  more  fanciful  than  real.  Of  course,  it  must  be  remem- 


24  HOW  TO  TEACH  AEITHMETIC 

bered  that  these  tests  were  given  to  upper  sixth  grade 
children  only.  It  seems  quite  probable  that  children  of 
less  maturity  and  training  would  find  columns  of  different 
lengths  varying  in  difficulty.  This  is  one  of  the  simple 
problems  awaiting  solution. 

Arithmetic  and  Formal  Discipline 

Practically  every  scientific  study  in  the  field  of  educa- 
tion has  shed  more  or  less  light  on  the  mooted  question  of 
formal  discipline.  Although  men  familiar  with  the  changed 
point  of  view  have  grown  increasingly  careful  in  their  dis- 
cussions of  the  educational  value  of  the  various  subjects 
of  study,  still  such  terms  as  arithmetical  ability,  historical 
ability,  and  literary  ability  are  current.  To  find  out 
whether  such  a  general  ability  exists  in  arithmetic  was  one 
of  the  problems  of  Dr.  Stone.  Logically,  when  one  uses 
such  terms  as  arithmetical  ability  or  historical  ability,  he 
presupposes  that  but  one  ability  exists  for  each  of  these  sub- 
jects, that  there  is  no  such  thing  as  a  plurality  of  abili- 
ties. This  conception  harks  back  to  the  days  of  faculty 
psychology,  which  considered  the  human  mind  as  made  up 
of  separate  faculties,  each  of  which  was  located  in  a  sep- 
arate compartment  of  the  brain  and  could  be  trained  by  a 
single  subject  of  study.  For  example,  the  faculty  of  rea- 
soning could  be  trained  best  by  the  study  of  mathematics, 
memory  by  the  study  of  language,  and  observation  by 
nature  study  and  science.  This  made  education  a  rela- 
tively simple  matter.  How  many  subjects  should  the  cur- 
riculum consist  of  was  answered  for  the  educationist  of  that 
day:  just  as  many  as  there  were  faculties  of  the  mind — 
no  more  and  no  less.  The  advocates  of  formal  discipline 
went  one  step  farther:  they  held  that  skill  in  one  field 
meant  that  this  skill  was  correspondingly  serviceable  in 


SCIENTIFIC  STUDIES  IN  ARITHMETIC  25 

every  field,  that  the  training  in  reasoning  one  got  from 
the  study  of  mathematics  made  him  equally  good  in  rea- 
soning everywhere.  The  training  in  memory  secured  from 
the  study  of  languages  meant  an  equally  tenacious  memory 
in  every  field.  In  other  words,  the  training  and  skill  which 
one  secured  in  any  special  field  could  be  generalized 
and  applied  to  every  field.  A  good  reasoner  in  mathe- 
matics would,  therefore,  be  equally  good  in  reasoning  in 
metaphysics  and  theology ;  a  good  memory  for  poetry 
would  likewise  mean  an  equally  strong  memory  for  faces, 
names,  dates  in  history,  or  scores  in  games.  This  effect  of 
special  training  was  another  concept  which  made  education 
a  simple  matter.  Only  a  few  subjects  were  needed, — just 
as  many  as  there  were  faculties  of  the  mind,  and  each  of 
these  had  the  special  and  unique  function  of  training  some 
particular  ability  which  in  some  mysterious  way  spread 
itself  and  made  unnecessary  the  development  of  this  ability 
in  other  fields.  A  few  things  prepared  one  for  success  in 
any  walk  of  life.  It  was  a  beautiful  scheme.  The  value  of 
every  subject  was  interpreted  in  terms  of  its  mind-train- 
ing value. 

Experience  has  furnished  us  with  numerous  instances  of 
the  failure  of  the  plan  to  work.  Modern  psychology  with 
its  functional  point  of  view  has  corroborated  these  common 
sense  experiences  by  demonstrating  conclusively  that  our 
mental  life  consists  of  abilities  and  not  of  faculties,  and 
that  these  abilities  become  specialized.  Training  in  one 
field  does  not  necessarily  transfer  at  all  to  other  fields. 
This  is  particularly  true  if  the  fields  are  dissimilar.  That 
doctrine,  therefore,  which  calls  for  the  establishment  of 
generalized  habits  is  a  psychological  absurdity.  Habits  are 
specific  responses  to  specific  stimuli.  Instead  of  there  being 
a  faculty  of  attention,  a  faculty  of  memory,  a  faculty  of 
reasoning,  there  are  abilities  for  attending,  for  memoriz- 


26  HOW  TO  TEACH  AEITHMETIC 

ing,  and  for  reasoning,  each  of  which  must  be  trained 
through  responses  to  a  special  type  of  stimulus.  The  result 
is  that  we  have  those  habits  of  attending,  of  memorizing, 
and  of  reasoning,  which  correspond  to  the  special  fields  in 
which  they  have  been  trained.  This  means  that  we  must 
teach  each  fact  or  theory  worth  teaching  as  if  the  salvation 
of  the  intellectual  world  depended  upon  it,  for  it  may  be 
that  the  limited  training  one  gets  from  any  one  of  them  will 
fail  to  modify  us  in  some  other  desirable  way.  This  means 
that  a  newer  and  heavier  obligation  rests  upon  the  teachers 
of  to-day. 

However,  we  must  not  presume  that  our  educational 
forefathers  were  altogether  mistaken.  Scientific  studies 
have  shown  that  training  in  one  field  is  serviceable  in 
another  in  proportion  as  the  two  fields  are  identical  in  sub- 
ject matter  or  in  method,  or  as  the  ideals  of  work  gathered 
in  one  are  useable  in  the  other.  As  many  different  types 
of  training  are  needed  as  there  are  fields  to  which  students 
should  be  adjusted. 

Dr.  Stone's  thesis  had  certain  important  contributions 
to  make  to  this  problem :  These  are  shown  in  the 

1.  Ratings  of  Cities 

The  various  school  systems  measured  show  a  great  lack 
of  uniformity  in  both  fundamentals  and  reasoning.  The 
order  of  the  cities  when  distributed  for  fundamentals  is 
not  the  same  as  when  distributed  for  reasoning.  Not  only 
does  the  order  differ  when  these  two  types  of  arithmetical 
processes  are  compared,  but  it  differs  when  one  fundamental 
is  compared  with  another.  For  example,  City  XX,  choos- 
ing at  random,  stood  lowest  in  addition,  next  to  lowest  in 
subtraction,  third  from  lowest  in  multiplication,  and  fourth 
from  lowest  in  division.  Now,  "if  the  net  result  of  arith- 


SCIENTIFIC  STUDIES  IN  ARITHMETIC  27 

metic  were  a  product,  each  system  would  have  the  same 
relative  position  in  each  phase  of  the  subject."     (p.  24) 

The  ratings  of  cities  with  reference,  to  the  mistakes  made, 
naturally  show  the  same  thing ;  i.  e.,  a  plurality  of  abilities. 
The  twenty-six  systems  measured  range  in  mistakes  made 
from  14.4  per  cent,  to  45.1  per  cent.  The  relative  rankings 
of  the  cities  as  to  mistakes  show  almost  no  uniformity  at 
all.  Cities  are  not  only  unlike  in  their  mathematical 
achievement,  but  in  their  lack  of  achievement  as  well. 

2.  Achievements  of  Pupils  as  Individuals 

When  individuals  are  compared  as  to  their  ability  along 
any  of  these  lines,  the  wide  variability  of  their  achievement 
becomes  obvious.  This  is  true  whether  we  compare  indi- 
viduals in  different  systems  or  individuals  in  the  same  sys- 
tem. This  difference  in  attainment  must  be  due  more  to 
differences  in  original  nature  than  to  differences  in  the 
course  of  study  or  in  the  character  of  the  instruction 
pupils  receive. 

School  systems  and  individuals  differ  less  widely  in 
their  achievements  in  fundamentals  than  in  reasoning.  This 
may  be  due  to  the  fact  that  the  fundamentals  are  better 
taught,  that  we  know  more  about  the  psychology  of  habit 
formation  than  we  do  about  the  psychology  of  reasoning. 
It  seems  true  that  if  a  class  is  drilled  upon  any  one  of  the 
fundamental  operations  the  variability  decreases,  whereas 
if  it  is  trained  on  problems  in  reasoning  the  variability 
increases.  In  other  words,  we  are  more  alike,  or  may  be 
made  more  alike,  as  to  our  habits  in  any  ability  in  arith- 
metic than  we  are,  or  can  be,  as  to  our  abilities  in 
reasoning. 

Dr.  Stone  made  a  comparison  of  the  attainments  of  boys 
and  girls  with  the  result  that  there  was  no  evidence  to 


28  HOW  TO  TEACH  ARITHMETIC 

show  chat  girls  are  either  more  or  less  stupid  than  boys 
in  arithmetic. 

3.  Relationship  of  Abilities 

It  has  already  been  noted  that  systems  and  individuals 
do  not  occupy  the  same  stations  in  different  traits  in 
arithmetic.  If  they  did,  then  the  relationship  would  be 
perfect;  a  person,  then,  would  do  equally  well  in  each 
phase  of  arithmetic, — a  certain  degree  of  ability  in  fun- 
damentals would  call  for  a  corresponding  degree  of  ability 
in  reasoning.  The  extent  to  which  kinship  exists  be- 
tween abilities,  the  extent  to  which  power  is  transferred 
from  one  ability  to  another,  is  measured  by  "the  coefficient 
of  correlation. ' '  The  coefficient  of  correlation  is  "a  single 
figure  so  calculated  from  the  individual  records  as  to  give 
the  degree  of  relationship  between  the  two  traits  which 
will  best  account  for  the  separate  cases  in  the  group.  In 
other  words  it  expresses  the  degree  of  relationship  from 
which  the  actual  cases  might  have  arisen  with  the  least 
improbability.  It  has  possible  values  from  4~  100  per  cent 
through  0  to  — 100  per  cent.  (Quoted  from  Thorndike's 
" Educational  Psychology/'  p.  25)  "A  coefficient  of  cor- 
relation between  two  abilities  of  +  100  Per  cent  would 
mean  that  the  best  system  or  pupil  in  the  group  in  one 
ability  would  be  the  best  in  the  other;  that  the  worst 
system  or  pupil  in  the  one  would  be  the  worst  in  the  other ; 
that  if  the  individuals  were  arranged  in  order  of  excellence 
in  the  first  ability  and  then  in  the  order  of  excellence  in 
the  second,  the  two  rankings  would  be  identical,  that  the 
station  of  any  pupil  in  one  would  be  identical  with  his  sta- 
tion in  the  other.  A  coefficient  of  --  100  per  cent  would,  per 
contra,  mean  that  the  best  system  or  pupil  in  the  one  ability 
would  be  the  worst  in  the  other,  that  any  degree  of 
superiority,  would  go  with  an  equal  degree  of  inferiority 


SCIENTIFIC  STUDIES  IN  AEITHMETIC  29 

in  the  other,  and  vice  versa. "  (Stone,  p.  37.)  A  coefficient 
of  +  62  per  cent  would  mean  that  any  given  station 
in  the  one  trait  would  imply  62  hundredths  of  that  station 
in  the  other.  A  coefficient  of  -  -  62  per  cent  would  mean 
that  any  degree  of  superiority  would  involve  62  hun- 
dredths as  much  inferiority,  and  vice  versa.1 

From  what  has 'already  been  said  it  is  reasonably  clear 
that  equality  of  achievement  is  not  secured.  The  coefficients 
of  correlation  between  reasoning  and  the  fundamentals  bear 
this  out.  The  twenty-six  systems  when  related  show  the 
following  correlations  : 

Addition  with  subtraction 92 

Addition  with  multiplication 95 

Addition  with  division 90 

Subtraction  with  multiplication 95 

Subtraction  with  division 93 

Multiplication  with  division 92 

This  high  correlation  between  the  fundamentals  is  what 
we  expect.  It  is  of  interest  to  note  that  the  lowest  rela- 
tionship is  between  addition  and  division,  and  that  the 
relationship  between  addition  and  multiplication  is  just 
the  equal  of  that  between  subtraction  and  multiplication. 
Comparing  the  achievements  of  500  individuals  selected 
at  random,  Dr.  Stone  found  the  relationships  between 
reasoning  and  the  fundamentals  to  be  as  follows : 

Reasoning  with  all  the  fundamentals 32 

Reasoning  with  addition 28 

Reasoning  with  subtraction 32 

Reasoning  with  multiplication 34 

Reasoning  wTith  division 36 

i  The  statistical  technique  involved  in  the  calculation  of  these 
coefficients  may  be  found  in  Thorndike's  "  Mental  and  Social 
Measurements."  The  Science  Press,  New  York  City. 


30  HOW  TO  TEACH  ARITHMETIC 

The  highest  correlation,  it  will  be  noted,  is  between 
reasoning  and  division.  This  probably  means  that  ability 
in  division  is  a  better  measure  and  criterion  of  ability  to 
reason  than  is  ability  in  any  other  fundamental.  If  one 
could  choose  only  one  fundamental  to  discover  whether 
children  could  reason  or  not,  he  should  choose  division. 

The  achievements  of  these  500  individuals  in  the  vari- 
ous fundamentals  were  related  as  follows: 

Addition  with  subtraction 50 

Addition  with  multiplication 65 

Addition  with  division 56 

Subtraction  with  multiplication 89 

Subtraction  with  division 95 

Multiplication  with  division 95 

When  compared  with  the  table  showing  the  relationships 
between  systems  as  to  fundamentals  it  is  clear  that  indi- 
viduals are  more  unlike  as  to  their  abilities  than  systems 
are  as  to  their  results.  This  table  also  lends  weight  to  the 
hypothesis  that  the  increasing  kinship  as  shown  by  the  cor- 
relations is  due  to  the  increase  in  the  amount  of  reason- 
ing involved.  ' i  It  seems  safe  to  say  tentatively  of  the  fun- 
damentals that  the  possession  of  ability  in  addition  is  the 
least  guarantee  of  the  possession  of  ability  in  others;  that 
the  possession  of  ability  in  multiplication  is  the  best  guar-. 
antee  of  the  possession  in  others;  and  that  this  probably 
means  that  multiplication  is  like  addition  on  its  mechanical 
side  and  like  division  on  its  thinking  side.  Hence,  if  it  is 
desired  to  measure  abilities  in  fundamentals  by  a  single 
test,  one  in  multiplication  would  be  best;  and  a  test  in 
division  would  probably  be  the  best  single  measure  of 
arithmetical  ability."  Stone,  p.  42. 

This  array  of  facts  furnishes  sufficiently  conclusive  evi- 
dence for  the  statement  that  there  is  no  such  thing  as  a 


SCIENTIFIC  STUDIES  IN  ARITHMETIC  31 

product  in  arithmetic, — what  we  get  is  a  complex  of  prod- 
ucts. There  is  no  such  thing  as  general  arithmetical 
ability, — there  are  many  arithmetical  abilities. 

Other  Evidence 

Dr.  Stone's  conclusion  has  been  corroborated  by 
several  other  studies.  Professor  Daniel  Starch,1  of  the 
University  of  Wisconsin,  conducted  a  somewhat  similar 
investigation.  "  Eight  people  practiced  fourteen  days  on 
oral  multiplication.  Before  and  after  practice  they  were 
given  six  tests  in  arithmetical  operations,  and  two  in  audi- 
tory memory  span.  For  comparison  seven  other  observers 
were  given  the  preliminary  and  final  tests  without  the 
practice  series.  The  practiced  observers  showed  from 
twenty  to  forty  per  cent  more  improvement  in  the  arith- 
metical tests  than  the  unpracticed  observers. "  The  sub- 
jects were  required  to  record  their  introspections.  These 
seemed  to  show  that  improvement  was  due,  not  to  an 
increase  in  the  memory  span,  but  to  an  ability  to  keep  the 
numbers  better  in  mind.  "The  improvement  in  the  end 
tests  was  due,  therefore,  to  the  identical  elements  acquired 
in  the  training  series  and  directly  utilized  in  other  arith- 
metical operations.  The  two  main  factors  were  (a)  the 
increased  ability  to  apprehend  and  hold  numbers  in  mind, 
and  (b)  the  acquisition  of  the  ability  to  utilize  arithmetical 
operations. ' ' 

Supervision  and  Results  in  Arithmetic 

A  practical  question  for  every  community  is,  "Does 
supervision  pay?"  As  the  field  of  education  is  differen- 
tiated, supervision  by  various  specialists  becomes  more 

i" Transfer  of  Training  in  Arithmetical  Operations."  Jour,  of 
Ed.  Psych.  2 :  306-310. 


32  HOW  TO  TEACH  AEITHMETIC 

and  more  imperative.  Apparently  some  one  whose  vision 
reaches  to  every  corner  of  the  field  is  needed,  that  the 
claims  and  demands  of  the  department  heads  of  a  school 
may  be  organized  for  effective  work.  School  organization 
is  one  of  the  essential  prerogatives  of  the  school  adminis- 
trator. Does  the  work  of  this  educational  engineer  modify 
achievement  in  any  particular  subject?  If  so,  how  much 
and  in  what  ways?  As  yet  no  satisfactory  answer,  based 
upon  trustworthy  experimental  studies,  has  been  secured 
to  these  questions.  The  twenty-six  school  systems  which 
we  have  been  describing  were  checked  with  reference  to 
whether  the  superintendent  alone  supervised  the  work  in 
arithmetic,  the  principal  alone,  or  the  principal  and  super- 
intendent combined.  Those  schools  that  ranked  best  were 
examined  by  both  of  these  officers ;  the  schools  that  ranked 
next  best  were  tested,  supervised,  and  inspected  by  the 
superintendent  alone;  and  those  that  ranked  poorest  by 
the  principal  alone. 

Time  in  Relation  to  Results  in  Arithmetic 

Our  school  programs  have  been  arranged  in  part  to 
correspond  to  the  work  curves  of  children.  In  some  books 
on  school  hygiene  it  is  claimed  that  the  work  curve  of 
children  rises  rapidly  during  the  forenoon,  reaching  its 
highest  mark  of  the  day  between  ten  and  eleven  o'clock; 
that  it  descends  until  after  the  noon  hour,  when  it  again 
rises,  reaching  its  high  mark  between  two  and  three  o  'clock, 
after  which  it  gradually  descends  until  bed  time.  Such  diffi- 
cult subjects  as  arithmetic  and  grammar  were  placed  on 
the  program  at  those  hours  of  the  day  when  the  energy 
of  the  children  was  supposed  to  be  at  its  highest  pitch.  * 
Subjects  like  music  and  drawing  were  taught  when  the 
children's  energy  was  at  ebb.  This  theory,  for  it  was  no 
more  than  that,  was  widely  taught  and  applied.  Dr. 


SCIENTIFIC  STUDIES  IN  ARITHMETIC  33 

Stone  has  given  us  some  new  light  upon  the  validity  of 
the  theory.  He  gave  his  tests  at  different  hours  of  the 
school  day.  The  results  showed  that  the  children  did  as 
well  one  time  as  another.  In  this  he  corroborated  the 
conclusions  reached  by  Mr.  Eice.  If  teachers  fail  to  get 
as  good  results  one  hour  of  the  day  as  another  in  arith- 
metic, the  failure  is  probably  no  criticism  upon  the  chil- 
dren. The  failure  is  more  probably  due  to  the  inability 
of  the  teacher  to  supply  adequate  motives.  The  feeling 
of  incompetency  may  be  broken  through  under  the  pres- 
sure of  strong  motives  and  new  intellectual  levels  reached. 
As  far  as  possible  a  teacher  should  be  permitted  to  arrange 
her  program  so  that  she  has  each  subject  coming  at  the- 
time  of  day  when  she  thinks  she  can  teach  it  best.  Then 
if  she  fails  the  criticism  of  her  work  may  with  justice  be 
far  more  severe  than  if  she  arranges  a  program  to  cor- 
respond to  hypothetical  work  curves. 

What  is  the  relation  of  the  total  amount  of  time 
expended  in  arithmetic  to  efficiency  in  the  subject?  If 
one  school  gives  one  hundred  minutes  a  week,  another 
two  hundred,  and  another  three  hundred,  may  we  expect 
the  results  in  the  second  school  to  be  twice  as  good  as  in 
the  first,  and  the  results  in  the  third  school  to  be  three 
times  as  good  as  those  in  the  first?  The  fact  is  the  thir- 
teen schools  that  received  less  than  the  median  time  cost 
did  slightly  better  than  those  that  received  more  than 
the  median  time  cost.  In  other  words,  there  was  no  direct 
ratio  between  time  expenditures  and  abilities.  "A  large 
amount  of  time  spent  on  arithmetic  is  no  guarantee  of  a 
high  degree  of  efficiency. "  (p.  62.)  The  chances  are 
exactly  even  that  if  one  were  to  choose  at  random  a  school 
having  less  than  the  median,  or  middle,  time  cost,  it  would 
stand  among  the  leaders  in  arithmetical  achievement,  and 
the  chances  are  even  that  if  one  were  to  choose  a  city 


34  HOW  TO  TEACH  ARITHMETIC 

with  more  than  the  middle  time  cost  it  would  rank  among 
the  poorest  in  arithmetical  achievement. 

The  best  results  were  secured  in  those  cities  that  had 
their  time  most  evenly  distributed,  grade  by  grade.  Mr. 
Rice  found  that  home  study  did  not  help  matters.  Dr. 
Stone  said  that  those  systems  that  required  home  work 
got  better  results  than  those  that  did  not.  This  is  a 
problem  demanding  further  investigation. 

Relation  of  Course  of  Study  to  Arithmetic 

A  good  course  of  study  does  not  guarantee  ability  in 
arithmetic.  Some  cities  with  excellent  courses  of  study 
were  among  the  poorest  in  results,  while  some  cities  with 
poor  courses  of  study  were  among  the  best  in  results.  It 
seems  that  ability  in  fundamentals  is  more  closely  related 
to  the  course  of  study  than  ability  in  reasoning. 

If  efficiency  in  arithmetic  cannot  be  measured  in  terms 
of  the  time  of  day  at  which  the  subject  is  taught,  the 
amount  of  time  expended  upon  it,  or  the  general  excel- 
lence of  the  course  of  study,  what  is  its  determining  cause  ? 
All  these  things  are  factors,  but  the  most  important  fac- 
tor, aside  from  supervision,  is  the  teacher  who  can  breathe 
personality  into  the  dead  materials,  and  who  can  transmute 
them  into  the  consciousness  of  the  children.  How  this  can 
best  be  done  is  still  largely  an  unsolved  problem.  It  seems 
that  there  must  be  many  ways  in  which  the  lifeless  mate- 
rials of  arithmetic  can  be  infused  with  life.  No  adequate 
survey  of  them  is  in  print. 

Standards  in  Arithmetic 

The  most  comprehensive  as  well  as  the  most  significant 
attempt  to  standardize  achievements  in  arithmetic  is  that 
of  Mr.  S.  A.  Courtis,  of  Detroit,  Michigan.  Mr.  Courtis 


SCIENTIFIC  STUDIES  IN  ARITHMETIC  35 

evolved  from  Stone's  study  the  idea  of  preparing  a  series 
of  tests  by  which  one  could  measure  with  considerable 
exactness  the  development  of  arithmetical  ability  through 
the  school  and  the  actual  attainment  of  any  individual  at 
any  time.  This  is  the  kind  of  tool  every  progressive 
teacher  and  efficient  supervisor  has  been  yearning  for; 
and  now  that  it  has  been  established  after  a  tremendous 
amount  of  patient  labor,  its  value  depends  upon  the  appli- 
cation that  is  made  of  it. 

The  tests  prepared  by  Mr.  Courtis  are  serviceable  for 
both  teachers  and  supervisors.  Pupils  also  can  use  them 
for  comparing  their  ability  at  one  time  with  their  ability 
at  another  time.  No  high  degree  of  technical  skill  or 
elaborate  knowledge  of  arithmetic  is  necessary  for  their 
application. 

Mr.  Courtis  limited  his  tests  to  simple  exercises  in  the 
four  fundamental  operations  and  to  one-  and  two-step 
problems'  in  reasoning.  The  tests,  eight  in  number,  are 
as  follows: 

Test  No.  1.    Addition  "1 

Test  No.  2.     Subtraction  ~     ,.     ,.        Aft 

m    XTVT  TX/T  i  •  T    A-        r  Combinations  0-9. 

Test  No.  3.     Multiplication 

Test  No.  4.     Division 

Test  No.  5.     Copying  figures    (rate  of  motor  activity) 

Test  No.  6.  Speed  reasoning  (simple  one-step  prob- 
lems) 

Test  No.  7.  Fundamentals  (abstract  examples  in  the 
four  operations) 

Test  No.  8.     Reasoning  (two-step  problems) 

To  measure  the  various  arithmetical  abilities  represented 
by  these  tests  grade  by  grade,  the  tests  must  be  given  in 
all  grades  under  exactly  similar  conditions.  The  tests  are 
printed  on  separate  sheets  of  paper;  the  pupils  turn  these 


36  HOW  TO  TEACH  AEITHMETIC 

face-up  at  a  given  signal;  and  then  work  at  full  speed 
under  a  time  limit.  There  are  more  problems  in  each  test 
than  any  pupil  can  solve  in  the  time  given.  By  this 
means  the  teacher  gets  not  only  the  range  and  central 
tendencies  of  each  ability  measured,  but  she  has  each 
individual  station  in  the  entire  distribution.  Room  can 
thus  be  compared  with  room,  grade  with  grade,  age  with 
age,  sex  with  sex,  one  individual  with  another.  If  the 
results  secured  for  a  given  ability  are  secured  by  different 
methods  and  if  the  pupils  in  two  rooms  are  of  about 
equal  ability,  then  the  tests  give  a  measure  of  the  value 
of  the  methods.  If  the  methods  are  the  same  and  the 
children  of  two  rooms  of  equal  ability,  then  the  results 
measure  the  efficiency  of  the  teaching.  By  the  use  of 
these  tests  any  careful  and  able  investigator  could  answer 
once  for  all  the  mooted  question  of  whether  children  in 
rural  schools  are  better  in  arithmetic  than  town  and  city 
children.  An  unpublished  study  by  Dr.  E.  H.  Taylor, 
of  the  Eastern  State  Normal  School  at  Charleston,  Illinois, 
shows  that,  contrary  to  expectation,  children  in  the  city 
schools  of  Charleston  and  Mattoon,  Illinois,  and  in  the 
training  school  tff  the  Normal  School,  without  exception 
averaged  higher  in  the  fundamental  arithmetical  opera- 
tions than  did  the  pupils  in  the  corresponding  grades  in 
the  rural  schools  of  the  county.  It  is  hardly  fair  to 
assume  that  Dr.  Taylor's  results  are  typical  for  the  United 
States,  and  yet  the  authors  feel  that  it  is  safer  to  accept 
them  than  to  accept  judgments  based  upon  crude  opinion. 
Whenever  experimental  evidence  is  weighed  against  mere 
opinion,  the  burden  is  upon  those  who  hold  the  opinion 
to  disprove  the  facts  arrived  at  through  experiment. 

In  confirmation  of  Stone's  thesis,  Mr.  Courtis  found 
that  school  systems  differ  less  widely  than  the  individuals 
within  a  system.  This,  Mr.  Courtis  believes,  "can  only 


SCIENTIFIC  STUDIES  IN  ARITHMETIC 


37 


mean  that  the  differences  in  the  abilities  of  individual 
children  are  greater  factors  in  determining  relative  rank 
in  school  work  than  all  the  differences  in  abilities  of 
teachers,  courses  of  study,  or  methods  of  work  combined." 
This,  of  course,  is  a  matter  which  can  be  settled  by  an 
extended  investigation.  The  true  answer  will  always  be  a 
matter  of  conjecture  until  children  are  taught  by  the 
same  methods  and  by  teachers  of  equal  ability.  Per- 
haps we  could  get  no  nearer  to  the  correct  answer  than 
to  give  the  tests  to  children  who  have  been  for  a  con- 
siderable number  of  years  in  an  orphan  asylum ;  there 
the  educative  environment  is  perhaps  more  uniform  than 
in  any  public  school  system. 

Mr.  Courtis  found  that  the  grades  overlapped  greatly 
in  each  of  the  abilities  measured.  This  fact  is  clearly 
shown  in  the  table  given  below. 

GENERAL  AVERAGE  FROM  TOTAL  DISTRIBUTIONS 


School 
^  Grade  No. 

Average  ol 
d  scores  for 
d  each  test 

Test 
No.  1 

6 

Test 
No.  2 

6 

Test  Test 
No.  3  No.  4 

Test 
No.  5 

29 

Test 
No.  6 

(*)   (•;•) 

Test 
No.  7 
(*)    (t) 

Test 
No.  8 
(*)    (t) 

9 

75 

21 

12 

10 

12 

51 

— 











3 

525 

26 

19 

16 

11 

63 

2.8 

2.1 

5.4 

1.7 

2.7 

0.6 

4 

1222 

33 

25 

23 

21 

70 

3.7 

2.5 

6.6 

3.6 

2.6 

0.8 

5 

1177 

40 

32 

30 

28 

80 

4.4 

3.4 

9.0 

5.3 

2.8 

1.2 

6 

1282 

46 

37 

34 

35 

88 

5.1 

4.4 

10.3 

6.9 

3.4 

1.7 

7 

1432 

51 

40 

38 

38 

98 

5.9 

5.2 

11.5 

7.6 

3.7 

2.2 

8 

1370 

57 

45 

43 

44 

102 

6.8 

6.1 

13.1 

8.9 

4.1 

2.7 

9 

412 

59 

47 

45 

47 

108 

6.9 

6.4 

13.7 

9.5 

4.1 

3.1 

10 

216 

57 

45 

43 

46 

112 

7.2 

6.7 

14.0 

9.5 

4.1 

3.1 

11 

151 

59 

47 

44 

48 

114 

7.9 

7.4 

14.4 

9.4 

4.5 

3.3 

12 

169 

61 

48 

44 

49 

112 

7.7 

7.2 

14.9 

10.8 

4.6 

3.6 

13 

462 

71 

56 

50 

56 

116 

8.6 

8.2 

16.8 

12.6 

5.3 

4.0 

14 

131 

74 

51 

58 

59 

124 

9.7 

9.1 

17.2 

11.8 

5.4 

4.1 

8679 


'T  Average  number  problems  attempted,     f  Average  number  right. 


38  HOW  TO  TEACH  ABITHMETIC 

"This  table  means,  when  put  into  words,  that  in  the 
third  grade,  for  example,  there  are  525  individual  scores 
in  each  test,  or  about  525.  If  in  some  test  there  were 
more,  as  for  example,  addition,  there  were  enough  less  in 
division  to  make  the  average  number  of  tests  525.  In 
addition  tests  it  was  found  that  the  standard  or  average 
ability  was  that  exhibited  by  a  child  who  could  record  26 
in  a  minute.  The  standard  in  subtraction  was  19  per 
minute,  in  multiplication  16,  division  11,  in  copying  fig- 
ures 63,  in  speed  reasoning  the  pupils  attempted,  on  the 
average,  2.8  problems  and  got  2.1  right,  and  so  on. 

Mr.  Courtis  holds  that  a  higher  score  than  the  average 
is  desirable  for  a  standard/  He,  therefore,  created  a 
standard  by  taking  the  lowest  score  of  the  best  30  per  cent 
of  the  children  in  each  of  the  eight  •  grades  measured. 
This  gives  the  following  table : 

STANDARD  SCORES 


T< 

Nc 

JSt 

).  1 

Test 
No.  2 

Test 

Nos.  3 
and  4 

Test 
No.  5 

Test 
No.  6 

Test 
No.  7 

Test 
No.  8 

Ats. 

Rt. 

Ats. 

Rt. 

Ats. 

Rt. 

Grade 

3 

26 

19 

16 

58 

2.7 

2.1 

5.0 

2.7 

2.0 

1.1 

Grade 

4 

34 

25 

23 

72 

3.7 

3.0 

7.0 

3.3 

2.6 

1.7 

Grade 

5 

42 

31 

30 

86 

4.8 

4.0 

9.0 

4.9 

3.1 

2.2 

Grade 

6 

50 

38 

37 

99 

5.8 

5.0 

11.0 

6.6 

3.7 

2.8 

Grade 

7 

58 

44 

44 

110 

6.8 

6.0 

13.0 

8.3 

4.2 

3.4 

Grade 

8 

63 

49 

49 

117 

7.8 

7.0 

14.4 

10.0 

4.8 

4.0 

Grade 

9 

65 

50 

50 

120 

8.6 

7.8 

15.0 

11.0 

5.0 

4.3 

Translating  this  table  into  words:  "At  the  end  of  a 
year's  careful  work  an  eighth  grade  child  should  be  able 
to  copy  figures  on  paper  at  the  rate  of  117  figures  per 
minute;  to  write  answers  to  the  multiplication  combina- 
tions at  the  rate  of  49  answers  per  minute ;  to  read 
simple  one-step  problems  of  approximately  30  words  in 
length  and  decide  upon  the  operation  to  be  used  in  their 
solution  at  the  rate  of  8  problems  a  minute  with  an 


SCIENTIFIC  STUDIES  IN  ARITHMETIC  39 

accuracy  of  90  per  cent;  to  work  abstract  examples  of 
approximately  10  figures  (twice  as  many  for  addition)  at 
the  rate  of  14.4  examples  in  10  minutes  with  an  accuracy 
of  70  per  cent;  to  solve  two-step  problems  of  approxi- 
mately 10  figures  at  the  rate  of  5  in  6  minutes  with  an 
accuracy  of  75  per  cent.  At  the  present  time  70  per  cent 
of  the  eighth  grade  children  cannot  meet  these  demands. 
But  it  must  be  borne  in  mind  that  three  per  cent  of  the 
fifth  grade  children  can,  and  that  experience  has  shown 
that  individual  care  and  a  little  well  arranged  drill 
produces  marked  changes  in  the  ability  of  most  children. 
Professor  Franklin  Bobbitt  translates  this  table  in  this 
way:  "In  simple  addition  operations,  the  third  grade 
teacher  should  bring  her  pupils  up  to  an  average  of  26 
correct  combinations  per  minute.  The"  fourth  grade 
teacher  has  the  task,  during  the  year  that  the  same  pupils 
are  under  her  care,  of  increasing  their  addition  speed 
from  an  average  of  26  combinations  per  minute  to  an 
average  of  34  combinations  per  minute.  If  she  does  not 
bring  them  up  to  the  standard  34,  she  has  failed  to  per- 
form her  duty  in  proportion  to  the  deficit ;  and  there  is 
no  responsibility  upon  her  for  carrying  them  beyond  the 
standard  of  34.  Her  task  is  simply  to  increase  their  addi- 
tion rate  from  26  to  34.  The  fifth  grade  teacher  is  to  take 
pupils  with  an  average  rate  of  34  and  bring  up  their 
speed  to  an  average  of  42,  a  perfectly  definite  task.  The 
sixth  grade  teacher  is  to  take  pupils  with  an  average  of 
42  and  to  carry  them  before  the  end  of  the  year  to  an 
average  of  50  combinations  per  minute.  The  seventh  grade 
teacher  increases  their  ability  from  50  combinations  to 
58.  The  eighth  grade  teacher  takes  them  with  58  com- 
binations per  minute  and  brings  them  up  to  63,  and  the 
ninth  grade  teacher  is  to  add  the  small  increment  of  2 
combinations  per  minute  during  the  ninth  grade.  In  like 


40  HOW  TO  TEACH  AEITHMETIC 

manner,  in  the  case  of  each  of  the  other  operations,  each 
teacher  has  his  own  special  increment  to  add  to  the  work 
of  his  predecessor  before  turning  his  partially  finished 
product  over  to  the  next  teacher  in  the  school.  This  table 
of  standard  scores  of  Mr.  Courtis  shows  us  the  ultimate 
standard  that  is  to  be  attained  at  the  end  of  the  school 
course,  and  it  also  shows  the  progressive  standards  to  be 
attained  at  each  stage  of  the  process  from  the  beginning 
to  the  end."1 

Reasoning  in  Arithmetic 

The  reasoning  ability  of  children  in  the  fourth,  fifth, 
and  sixth  grades  in  the  public  schools  of  Passaic,  New 
Jersey,  was  tested  by  Dr.  F.  G.  Bonser.2 

Perhaps  the  most  significant  single  contribution  made 
by  this  study  was  the  confirmation  of  President  G.  Stanley 
Hall's  theory  of  the  periodicity  of  the  learning  process. 
It  was  found  that  progress  is  not  regular,  but  that  there  is 
a  well  defined  rhythm.  The  nodes  or  crests  of  the  wave 
for  boys  appear  at  about  9  years  and  6  months,  at  about 
12  years,  and  at  about  14  years  and  6  months.  "For  the 
girls,  there  is  evident,  though  not  so  clearly,  a  rhythm  with 
its  crests  about  coincident  with  the  valleys  of  the  rhythm 
for  the  boys,  excepting  at  the  period  11  years  6  months 
to  12  years  6  months,  where  the  crests  become  nearly 
parallel.  In  so  far  as  these  tests  and  these  children  are 
typical,  then  we  can  predict  that  of  any  group  of  chil- 
dren of  these  grades,  ranked  on  the  basis  of  ability,  a  larger 
proportion  of  those  pupils  who  are  from  12  to  13  will  be 
found  in  the  highest  group  than  of  those  who  are  from 
11  to  12." 

1  Quoted  from  ' '  The  Twelfth  Yearbook  of  the  National  Society  for 
the  Study  of  Education.     Part  I,  pp.  21-22. " 

2  The  Eeasoning  Ability  of  Children.     Teachers'  College,  Columbia 
University,  Contributions  to  Education,  No.  37. 


SCIENTIFIC  STUDIES  IN  ARITHMETIC  41 

His  results  also  showed  that  the  greatest  gains  irrespec- 
tive of  sex  were  made  in  the  4A  and  5B  grades;  the 
smallest  gains  for  boys  in  5B  and  5 A  grades  and  for 
girls  in  the  6B  and  6A  grades.  His  results  also  confirmed 
the  opinion  that  those  of  superior  ability  were  the  youngest 
age  group.  Because  of  this  it  is  easily  possible  that  unless 
the  school  is  flexible  in  its  promotions  the  brightest  pupils 
may  be  the  most  seriously  retarded.  A  comparison  of  the 
sexes  showed  that  boys  are  a  little  better  than  girls  in 
reasoning  ability  in  arithmetic,  and  slightly  more  variable. 

The  studies  that  relate  more  specifically  to  practice  and 
practice  effects  will  be  discussed  in  the  chapter  on  Drill 
Work. 


PART  TWO 

CHAPTEE  III 

ACCURACY 

Prevalence  of  Inaccuracy 

A  common  criticism  of  the  schools  of  to-day  is  that  the 
pupils  have  been  permitted  to  become  lax  and  careless  in 
thought  and  in  expression.  The  modern  pupil  is  expected 
to  study  many  things  which  were  not  taught  in  the  schools 
of  the  last  generation,  but  there  is  truth  in  the  assertion 
that  no  small  part  of  his  knowledge  is  superficial  and 
inaccurate,  "a  collection  of  vague  ideas  rather  than  clear 
cut  notions  about  definite  things."  It  is  asserted  that 
while  the  pupils  of  to-day  can  think  and  write  on  more 
subjects  than  could  the  students  of  former  years,  their 
expressions  are  less  clear  and  coherent.  Teachers  should 
not  turn  a  deaf  ear  to  the  complaints  about  inaccuracy 
of  thought  and  statement.  The  complaints  come  from 
many  sources  and  seem  to  be  corroborated  by  overwhelm- 
ing evidence.  The  employer  maintains  that  it  is  difficult 
to  hire  a  boy  who  is  accurate  in  statement  and  who  has  a, 
mastery  of  even  the  four  fundamental  operations.  Teach- 
ers in  the  upper  grades  contend  that  the  pupils  who  enter 
their  classes  are  not  prepared  to  carry  the  work  because 
their  thinking  is  illogical  and  their  ability  in  computation 
is  poor. 

One  writer  asserts  that  all  the  complaints  about  inac- 
curacy are  an  evidence  that  accuracy  is  one  of  the  goals 
which  education  ought  to  reach.  Ex-President  Eliot 

43 


44  HOW  TO  TEACH  ARITHMETIC 

enumerates  among  the  essentials  of  the  cultured  mind  the 
ability  to  think  clear  and  straight.  Accuracy  is  one  of 
the  marks  of  the  scholar.  The  system  of  education  which 
minimizes  the  importance  of  accuracy  of  thought  and  of 
expression  is  relegating  to  a  subordinate  position  one  of 
the  essentials  of  true  scholarship  and  culture. 

« 
Accuracy  of  Thought 

If  there  is  one  subject  rather  than  another  in  the  cur- 
riculum which  should  be  characterized  by  a  high  degree 
of  accuracy,  that  subject  is  mathematics.  In  mathematics 
a  statement  is  either  right  or  it  is  wrong;  there  is  no 
middle  ground.  Mathematical  accuracy  is  proverbial.  The 
demand  of  the  present  is  for  greater  accuracy  in  both 
thought  and  computation.  Accuracy  of  thought  is  the 
more  fundamental  and  is  quite  frequently  the  basis  for 
accuracy  of  manipulation.  Myers  defines  accuracy  of 
thought  in  arithmetic  as  "the  degree  of  closeness  of 
expression  to  idea — the  adequacy  of  assertion  to  thought. '" 
As  the  pupil  matures,  his  accuracy  of  expression  should 
improve,  as  measured  by  an  absolute  standard,  for  his 
conception  of  quantitative  relationship  is  extended.  The 
expression  used  by  the  pupil,  often  exposes  the  mental 
process;  teachers  can  improve  accuracy  of  statement  by 
frequently  insisting  upon  full  and  concise  explanation. 

A  great  responsibility  rests  upon  the  teachers  of  the 
early  grades  for  securing  accuracy  of  expression  in  oral 
and  in  written  wort.  In  these  grades  inaccurate  habits 
•of  expression  are  often  formed  and  if  not  corrected  before 
the  pupil  reaches  the  upper  grammar  grades  they  are 
quite  likely  to  persist  and  to  interfere  seriously  with  the 
pupil's  progress  in  the  subject.  David  Eugene  Smith 
says,  "It  is  the  loose  manner  of  writing  out  solutions, 


ACCURACY  45 

tolerated  by  many  teachers,  that  gives  rise  to  half  the  mis- 
takes in  reasoning  which  vitiate  the  pupil's  work/'  and 
"teachers  are  coming  to  recognize  that  inaccuracies  of 
statement  tend  to  beget  inaccuracy  of  thought  and  so 
should  not  be  tolerated  in  the  schoolroom. "  There  is  real 
value  in  clear  cut  and  concise  statements  of  conditions  and 
arguments.  One  who  has  observed  recitations  in  a  large 
number  of  schools  is  impressed  by  the  looseness  and  care- 
lessness of  expression  in  both  oral  and  written  arithmetic. 
In  arithmetic,  as  in  other  subjects,  it  ordinarily  takes  at 
least  a  whole  sentence  to  express  a  thought. 

A  number  of  the  more  common  inaccurate  statements 
found  in  certain  text-books  and  frequently  permitted  to 
go  unchallenged  in  classrooms  will  now  be  considered. 
The  fact  that  certain  of  the  inaccuracies  considered  are 
found  in  text-books  of  superior  merit  does  not  lessen  the 
justness  of  the  criticisms  that  follow.  A  statement  is  not 
correct  merely  because  it  is  used  by  a  large  number  of 
authors.  The  question  at  issue  is  not  whether  this  or  that 
author  has  a  correct  statement  for  the  idea  under  con- 
sideration ;  we  are  not  considering  an  author  as  an  indi- 
vidual; "Not  who  is  right,  but  what  is  true,"  is  the  ques- 
tion at  issue.  The  good  teacher  will  be  open-minded  and 
candid  with  himself  and  will  consider  the  question  in  an 
impersonal  manner. 

If  enough  people  of  influence  habitually  use  an  inac- 
curate statement  until  its  "real"  meaning  becomes 
apparent  it  is  hardly  worth  while  to  raise  objection  to 
the  inaccuracy.  The  inaccuracies  which  follow  are  not  of 
this  type. 

Figures  and  Numbers 

One  common  inaccuracy  is  due  to  a  confusion  between 
the  symbol  and  that  which  the  symbol  represents.  Fre- 


46  HOW  TO  TEACH  AEITHMETIC 

quently  the  pupil  is  directed  by  the  author  or  the  teacher 
to  add  figures.  Figures  are  not  numbers;  they  are  the 
symbols  which  represent  numbers.  It  is  just  as  impossible 
to  add,  subtract,  multiply,  or  divide  figures  as  it  is  to 
harness  the  picture  of  a  horse.  The  figure  is  not  the  real 
thing;  it  is  that  which  represents  the  concept  to  the  mind. 
It  is  possible  to  perform  various  operations  upon  num- 
bers, but  these  operations  cannot  be  performed  upon  that 
which  merely  represents  numbers.  It  is  important  that 
both  teachers  and  pupils  discriminate  between  numbers  and 
the  characters  which  represent  them.  It  is  proper  to  direct 
the  pupil  to  add  the  numbers  represented.  The  teacher  is 
reminded  in  this  connection  that  accuracy  is  not  a  little 
thing,  but  even  in  little  things  we  should  be  accurate. 

Inaccuracy  in  Statement 

Such  statements  as  the  following  are  frequently  found 
in  classroom  work  and  in  some  text-books : 

3  +  4  =  7  +  5  =  12x2  =  24 

The  inaccuracies  in  the  above  statement  are  evident.  It 
is  not  true  that  3  +  4  =  7  +  5,  nor  does  7  +  5  =  12x2.  The  mere 
fact  that  the  correct  answer  is  obtained  does  not  justify  the 
inaccuracy  of  statement.  If  the  data  given  are  correctly 
used  and  all  the  computations  are  accurate,  the  answer  in 
any  problem  will  be  correct;  but  it  is  possible  to  obtain 
the  correct  answer  to  a  problem,  even  though  no  step  in  the 
solution  is  correct.  This  may  be  brought  about  by  a 
balancing  of  inaccuracies  in  computation.  The  process 
may  justify  the  answer,  but  the  answer  does  not  justify 
the  process. 

If  the  above  example  needs  solution,  it  should  be  put 
into  the  following  form:  3  +  4  =  7;  7  +  5  =  12;  12x2  =  24. 


ACCURACY  47 

Inaccuracy  in  the  Addition  of  Mixed  Numbers 
A  third  inaccuracy  is  the  following: 

I  =  T% 

i_  e  It  is  apparent  that  4§  does  not  equal  &  as  the  solu- 

tion asserts,  nor  does  3£  equal  &  or  2f  equal  &. 


The  solution  may  be  correctly  expressed  in  several  ways. 
The  following  does  not  violate  the  truth  in  any  of  its  state- 
ments : 


Sum 


Inaccuracies  in  Multiplication  and  Division 

The  definitions  of  multiplication  and  division  are  often 
violated. 

In  multiplication  the  multiplier  must  always  be  abstract, 
and  the  product  must  always  be  of  the  same  denomination 
as  the  multiplicand. 

In  division  the  quotient  is  always  of  the  same  name  as 
the  dividend  when  the  divisor  is  abstract. 

Such  inaccurate  statements  as  the  following  are  not 
uncommonly  made  in  classrooms: 

2x$50  =   100         $100-   4  =   25 

2x   50  =  $100  100-   4  =  $25 

$100-  $4  =  $25 

Feet  Multiplied  by  Feet 

The  following  is  an  inaccurate  expression  :  4  ft.  x  5  ft.  = 
20  sq.  ft.  The  first  factor  (4  ft.)  cannot  represent  the 


48  HOW  TO  TEACH  AEITHMETIC 

multiplier,  because  it  is  not  abstract;  for  the  same  reason 
the  second  factor  (5  ft.)  cannot  be  the  multiplier.  More- 
over, since  the  product  must  be  of  the  same  name  or 
denomination  as  the  multiplicand,  it  is  apparent  that  if 
the  result  (20  sq.  ft.)  is  correct,  neither  4  ft.  nor  5  ft.  can 
be  the  multiplicand,  for  they  are'  not  of  the  same  name  as 
the  product.  Since  the  problem  involves  neither  multipli- 
cand nor  multiplier,  it  is  evidently  not  a  problem  in 
multiplication.  This  multiplying  of  feet  by  feet  and  call- 
ing the  result  square  feet  is  one  of  the  most  common  in- 
accuracies of  statement  in  arithmetic.  If  a  rectangle  is 
4  ft.  wide  and  5  ft.  long,  we  may  find  its  area  by  the 
following  method :  4  x  5  x  1  sq.  f t.  =  20  sq.  ft. 


A 
M 

4 


D 


The  area  represented  by  ABCD  is 
4X5X1  sq.  ft.  because  it  contains  4 
rows  similar  (  to  AMHD  which  contains 
5  X  1  sq.  ft. 


The  inaccuracy  of  the  statement,  4  ft.  x  5  ft.  =  20  sq.  ft, 
may  also  be  shown  by  interpreting  multiplication  as  a  short 
method  for  addition.  It  is  evidently  impossible  to  add  4  ft. 
(or  5  ft.)  to  itself  any  number  of  times  and  get  a  result 
whose  denomination  is  square  feet.  It  would  be  possible 
to  extend  the  definition  of  multiplication  so  as  to  make 
4  ft.  x  5  ft.  =  20  sq.  ft.  a  correct  expression,  but  this  has 
not  been  done,  and  there  is  no  necessity  for  doing  so  in 
the  elementary  school. 

Common  Fractions 

In  the  analysis  of  problems  in  fractions  and  in  percent- 
age, inaccuracies  of  expression  are  very  common.  The 


ACCURACY  49 

following   illustration   will  indicate   the   nature   of  these 
inaccuracies : 

Problem.  Two-fifths  of  a  number  equals  12.  Find  the 
number. 

I  II 

AN  INCORRECT  SOLUTION  A  CORRECT  SOLUTION 

.|  =  the  number  £  of  the  number  =  the  number 

f  =  12  f  of  the  number  =  12 

i  =  i  of  12  =  6  i  of  the  number  =  $  of  12  =  6 

|  =  5  X  6  =  30  |  of  the  number  =  5  X  6  =  30 

Not  a  statement  in  solution  I  is  true.  If  f  equal  the 
number,  it  must  be  true  that  the  number  equals  1,  since  £ 
equal  1 ;  but  the  number  is  30,  and  not  1.  If  f  equals  12, 
then  f  must  equal  -%°-,  the  equivalent  of  12 ;  but  this  is 
evidently  absurd.  If  f  equals  12,  then  £  must  equal  6 ;  but 
6  equals  -^,  and  £  does  not  equal  \° .  The  last  statement  of 
solution  I  is  seen  to  be  absurd  as  soon  as  we  substitute  for 
f  its  equal,  1,  for  we  then  have  1  =  30.  To  say,  "Let  f  equal 
the  number, ' '  or  "  assume ' '  that  f  equals  the  number,  does 
not  eliminate  the  inaccuracy.  The  incorrectness  of  solu- 
tion I  is  not  an  incorrectness  in  reasoning  after  the  first 
steps  are  taken.  The  reasoning  is  faultless  after  the  first 
two  steps.  The  incorrectness  lies  in  the  inaccuracy  of 
statement.  The  first  two  statements  are  untrue,  hence  the 
reasoning  based  upon  them  is  untrue.  Any  absurd  con- 
clusion may  be  reached  by  correctly  reasoning  from  false 
premises. 

At  first  sight  it  appears  that  solution  II  does  not  differ 
greatly  from  solution  I,  but  a  brief  study  of  the  two  solu- 
tions will  convince  one  of  the  inaccuracy  of  the  first  and 
the  correctness  of  the  second.  The  word  "of"  when  used 
with  common  fractions  indicates  multiplication.  Since  $ 
is  a  common  fraction,  the  first  statement  of  solution  II  is 
equivalent  to  "1  times  the  number  equals  the  number." 


50  HOW  TO  TEACH  AEITHMETIC 

This  is  always  true.  Contrast  this  with  the  corresponding 
step  of  solution  I,  which  asserts  that  -|  (or  1)  equals  the 
number.  This  last  statement  is  never  true,  unless  the  re- 
quired number  is  actually  1.  The  second  statement  in  solu- 
tion II  is  given  in  the  original  problem,  and  hence  is  known 
to  be  true.  From  these  two  statements,  or  premises,  both  of 
which  are  true,  by  accurate  reasoning  we  arrive  at  the 
conclusion  which  is  true. 

Many  teachers  may  feel  that  their  pupils  do  not  have 
time  when  analyzing  problems  of  this  type  to  write  an 
analysis  similar  to  solution  II.  It  need  only  be  urged  that 
no  teacher  and  no  pupil  should  ever  be  so  pressed  for  time 
or  for  space  that  he  hasn't  time  to  tell  the  truth.  It  is  not 
correct  to  say  that  |  equals  A's  age,  or  A's  money,  or  the 
distance  from  B  to  C.  The  accurate  statements  are : 

f  of  A's  age  equals  A's  age. 

-§•  of  A's  money  equals  A's  money. 

f  of  distance  from  B  to  C  equals  distance  from  B  to  C. 
It  is  not  intended  in  the  preceding  illustrations  to  convey 
the  idea  that  the  suggested  forms  are  the  only  correct 
forms..  An  accurate  statement  of  facts  may  often  be  made 
in  a  variety  of  ways,  and  it  matters  little  how  the  fact  is 
stated,  provided  only  that  the  statement  is  accurate,  concise, 
clear,  and  grammatical. 

Inaccuracies  in  Percentage 

The  analysis  of  problems  in  percentage  often  leads  to 
inaccuracies  similar  to  the  one  just  considered.  "Forty 
per  cent  of  a  number  equals  80.  Find  the  number."  If 
we  begin  the  solution  by  asserting  that  100%  equals  the 
number,  we  are  evidently  committing  the  same  inaccuracy 
of  statement  that  was  pointed  out  above,  since  100%  equals 
,  and  it  is  not  correct  to  say  that  |$-§-  equal  the  number. 


ACCURACY  51 

The  correct  statement  is  100%  of  the  number  equals  the 
number. 

Mensuration 

It  is  incorrect  to  speak  of  the  square  root  of  49  square 
feet  as  7  feet.  By  the  square  root  of  a  number  is  meant 
that  which  multiplied  by  itself  produces  the  given  number. 
Thus,  the  square  root  of  25  is  5,  because  5x5  =  25.  If  the 
square  root  of  49  sq.  ft.  =  7  ft.,  it  follows  that  7  ft.  x  7  ft.  ~ 
49  sq.  ft. ;  but  from  the  argument  immediately  preceding, 
it  appears  that  this  is  impossible.  It  is  incorrect  to  say 
27  cu.  ft.  -r  9  sq.  ft.  =  3  ft.,  since  9  sq.  ft.  x  3  ft.  does  not 
equal  27  cu.  ft.  For  justification  of  this  point  and  for 
fuller  explanation,  see  the  chapter  on  Mensuration, 

Longitude  and  Time 

In  the  solution  of  problems  in  Longitude  and  Time, 
pupils  should  be  taught  that  15  degrees  correspond  to  1 
hour  of  time,  instead  of  15  degrees  equal  1  hour  of  time. 
The  relationship  is  not  one  of  equality  in  the  mathematical 
sense  of  the  word. 

The  preceding  inaccuracies  are  common  in  our  schools, 
and  they  are  all  in  direct  contradiction  to  the  habits  of 
accuracy  of  expression  which  should  be  cultivated  in  the 
study  of  arithmetic.  Moreover,  such  inaccuracies  as  those 
just  pointed  out  are  likely  to  lead  to  confusion  and  to 
inaccuracy  of  thought.  Teachers  should  not  tolerate  them 
in  their  classes.  Such  inaccuracies  of  expression  cause 
' '  many  teachers  to  sit  up  at  night  to  correct  mistakes  which 
they  had  better  sit  up  in  the  daytime  to  prevent. ' ' 

Inaccuracy  in  Computation 

Not  only  should  accuracy  of  statement  be  secured,  but 
a  high  degree  of  accuracy  of  computation  should  be  ac- 


52  HOW  TO  TEACH  AEITHMETIC 

quired.  The  evidence  of  inaccurate  computation  in  our 
schools  is  overwhelming.  It  is  not  unusual  to  find  a  group 
of  seniors  in  high  school  not  twenty-five  per  cent  of  whom 
can  add  ten  numbers  of  four  digits  each  and  secure  the 
correct  result  the  first  time. 

The  business  man  contends  that  it  is  very  difficult  to 
secure  a  boy  who  can  add,  subtract,  multiply,  and  divide 
with  speed  and  with  a  high  degree  of  accuracy. 

When  a  boy  or  girl  seeks  a  position  in  which  accur- 
acy and  rapidity  of  computation  are  essential,  it  is  of 
little  consequence  that  the  applicant  can  solve  problems 
and  explain  them  in  a  clear  and  concise  manner,  if  the 
ability  to  perform  the  arithmetical  operations  with  speed 
and  accuracy  has  not  been  developed.  In  many  vocations 
work  that  is  inaccurate  is  of  no  value  whatever. 

Accuracy  in  Practical  Work 

It  is  not  always  possible  to  secure  absolute  accuracy,  but 
the  standard  should  be  set  as  high  as  can  be  attained.  The 
Coast  and  Geodetic  Survey  uses  an  ice-bar  apparatus  for 
measuring  a  base  line.  An  ordinary  metal  bar  would  be 
subjected  to  expansion,  and  the  results  would  not  be  accu- 
rate. By  supporting  the  metal  measuring-bar  in  a  trough 
packed  in  ice,  it  is  maintained  at  a  uniform  temperature, 
and  a  base  line  can  be  measured  with  an  error  of  only  one 
part  in  2.5  million.  The  surveyor's  calculation  of  areas 
must  be  accurate,  and  so  he  checks  his  computations  by 
totals  of  latitudes  and  departures.  In  1900  the  United 
States  Coast  and  Geodetic  Survey  completed  the  measure- 
ment of  an  arc  along  the  39th  parallel  from  Cape  May  in 
New  Jersey  to  Point  Arenas  in  California,  a  distance  of 
2,625  miles.  So  carefully  was  the  work  done  that  the  total 
amount  of  probable  error  does  not  exceed  100  feet. 


ACCUEACY  53 

Many  pupils  go  through  the  entire  school  course  without 
appreciating  the  value  of  accuracy.  ' '  Arithmetic  as  a  tool  is 
almost  useless  unless  it  has  an  edge  keen  enough  to  do  its 
work  with  considerable  speed  and  absolute  accuracy.  Speed 
must  first  be  attained  by  having  pupils  deal  with  things  so 
simple  that  practically  all  the  attention  can  be  given  to  the 
speed  itself.  Teachers  permit  and  sometimes  encourage  in- 
accuracy by  giving  high  grades  for  the  correct  process,  even 
if  the  result  is  absurd.  Pupils  soon  form  the  habit  of  accu- 
racy when  they  find  that  inaccurate  results  are  always 
marked  zero."1  The  good  is  the  enemy  of  the  best,  in  the 
arithmetic  class  as  well  as  in  many  other  places.  It  is  not 
easy  to  secure  work  that  is  characterized  by  a  high  degree 
of  accuracy.  The  teacher  must  continually  insist  upon 
accuracy,  but  the  results  fully  justify  the  time  spent 
and  the  effort  expended.  After  a  high  degree  of  accuracy 
has  been  secured,  the  speed  can  be  gradually  increased,  but 
mere  practice  will  never  make  the  shiftless  and  careless 
pupil  accurate  in  his  computations.  With  such  a  pupil 
there  is  almost  an  inverse  relation  between  speed  and  accu- 
racy. The  faster  he  works,  the  greater  the  number  of 
mistakes  he  makes.  We  must  not  permit  our  pupils  to 
regard  accuracy  of  computation  as  a  matter  of  small  impor- 
tance. Some  of  the  pupils  of  our  schools  seem  to  believe 
that  accuracy  in  computation  is  of  so  little  moment  as  to 
be  almost  beneath  their  intellectual  dignity.  It  is  hard  to 
eradicate  such  an  idea  after  the  pupil  reaches  the  high 
school  age. 

Speed  and  Accuracy 

It  was  formerly  thought  that  there  is  a  very  direct  rela- 
tion between  accuracy  and  speed.  It  was  asserted  that  the 
most  rapid  computers  are  always  the  most  accurate.  The 

1N.  E.  A.  Proceedings,  1906,  p.  101. 


54  HOW  TO  TEACH  ARITHMETIC 

relation  is  not  so  intimate  as  was  formerly  supposed.  Stone1 
showed  that  there  is  not  a  necessary  relationship  between 
speed  and  accuracy.  In  his  investigation  the  school  system 
which  ranked  first  in  accuracy  was  fourth  in  rapidity, 
while  the  system  which  ranked  second  in  accuracy  was 
tenth  in  rapidity.  The  effect  of  drill  on  speed  and  upon 
accuracy,  as  indicated  by  .some  recent  investigations,  is 
considered  in  detail  in  the  chapter  on  Drill. 

The  subject  of  checks  is  closely  related  to  that  of  accu- 
racy of  computation.  Even  the  professional  mathematician, 
who  makes  computations  with  great  frequency,  does  not 
guarantee  the  accuracy  of  his  results  until  he  has  applied 
to  them  some  adequate  check.  Some  simple  checks  for 
accuracy  of  results  should  be  extensively  used  in  the 
schools,  and  a  number  of  such  checks  are  considered  in 
the  following  chapter. 

Copying  of  Figures 

Copying  figures  is  now  one  of  the  tests  in  civil  service 
examinations,  and  because  of  its  importance  it  should  have 
some  consideration  in  the  work  in  arithmetic.  Recently 
one  of  the  authors  investigated  a  group  of  seventh  grade 
pupils  with  respect  to  inaccuracy  in  copying  figures  from 
a  printed  page.  Mistakes  of  this  type  are  of  two  general 
kinds;  first,  the  inserting  of  figures  that  do  not  occur  in 
the  original  copy,  and,  second,  the  interchange  of  figures 
which  actually  do  occur.  As  an  example  of  the  first 
class  of  errors,  the  copying  of  59  for  86  might  be  cited, 
while  the  copying  of  34  for  43  exemplifies  the  second  class. 
Errors  of  the  first  class  were  more  frequent  than  those  of 
the  second  class,  in  the  ratio  of  28  to  11. 

1  Stone,  ' '  Arithmetical  Abilities  and  Some  Factors  Which  Deter- 
mine Them." 


ACCURACY  55 

Pupils  who  are  below  the  average  in  speed  miscopy 
figures  more  frequently  than  those  who  are  above  the 
average.  Figures  were  omitted  in  the  copying  about  one- 
third  as  frequently  as  they  were  miscopied. 

In  a  test  in  the  seventh  grade  involving  only  addition, 
subtraction,  multiplication,  and  division,  it  is  probable  that 
from  four  to  eight  per  cent  of  all  mistakes  are  due  to 
miscopying  or  omission  of  figures.  No  doubt  this  percentage 
is  higher  in  the  lower  grades. 


CHAPTER  IV 
CHECKS 

In  the  preceding  chapter  the  necessity  and  the  importance 
of  accuracy  of  thought  and  expression  were  considered.  The 
effect  of  drill  upon  accuracy  of  computation  is  considered 
in  the  chapter  on  Drill.  We  will  now  consider  some  other 
methods  by  which  increased  accuracy  of  computation  may 
be  secured. 

Necessity  of  Using  a  Check 

Long  practice  in  mathematical  computations  is  indispen- 
sable to  both  accuracy  and  speed,  but  practice  alone  is  not 
sufficient  to  produce  the  degree  of  accuracy  which  is  essen- 
tial in  mathematics.  Even  accountants  and  professional 
mathematicians  are  not  willing  to  guarantee  the  accuracy 
of  their  computations  until  some  appropriate  check  has 
been  applied.  The  surveyor  checks  his  long  chain  of  calcu- 
lation by  totals  of  latitudes  and  departures ;  the  navigator 
who  computes  his  position  at  sea  must  check  his  results  if  he 
would  guard  the  lives  and  the  cargo  entrusted  to  his  care. 
A  mistake  of  a  mile  in  computing  his  results  may  mean  the 
loss  of  hundreds  of  lives  and  of  thousands  of  dollars.  If 
a  result  will  check  it  is  correct,  if  it  will  not  check  it  is 
incorrect.  The  best  check  to  be  applied  is  determined  by 
a  number  of  factors.  The  check  should  vary  with  the 
process  and  with  the  maturity  of  the  pupil. 

56 


CHECKS  57 

Checking  by  the  Teacher 

Checks  are  of  many  kinds.  The  teacher  may  serve  as  a 
valuable  check  upon  the  work  of  the  pupil,  but  many 
teachers,  by  habitually  accepting  inaccurate  and  untidy 
work,  do  not  stimulate  the  pupil  to  greater  accuracy.  Fre- 
quently work  that  is  not  correct  in  every  particular  should 
be  considered  wholly  wrong,  and  the  teacher  who  habitually 
accepts  work  that  is  not  accurate  is,  perhaps  unconsciously, 
discounting  the  value  of  accuracy.  The  teacher  should 
always  be  a  check  upon  the  work  of  the  pupil,  but  the  check 
of  greatest  value  to  the  pupil  is  the  one  that  he  himself 

applies. 

Pupil  Must  Check  His  Results 

Unless  a  pupil  has  acquired  the  habit  of  checking  his 
results,  he  is  not  master  of  the  situation;  such  artificial 
devices  as  answer  books  are  of  value  chiefly  in  checking 
unusually  long  or  involved  problems.  The  earlier  the  con- 
stant verification  of  results  is  begun,  the  more  automatically 
it  will  be  practiced.  It  is  not  uncommon  for  teachers  to 
object  to  the  use  of  checks  on  the  ground  that  they  diminish 
the  number  of  problems  that  it  is  possible  for  a  pupil  to 
work.  The  fact  that  the  check  itself  is  a  problem  is  fre- 
quently overlooked.  The  essential  thing  is  to  gain  "well- 
grounded  self-confidence ' '  to  know  that  the  result  is  correct. 

The  checks  discussed  in  the  following  pages  are  used  by 
teachers  and  by  men  in  the  various  vocations  in  which 
accurate  computation  plays  an  important  part.  Not  all  of 
the  checks  should  be  taught  to  any  class  of  pupils.  One 
good  check,  for  a  given  process,  so  well  known  and  sc  fre- 
quently used  that  its  application  has  become  almost  auto- 
matic to  the  pupil  is  of  more  value  to  him  than  three  or 
four  checks  all  of  which  can  be  used  but  no  one  of  which 
has  been  thoroughly  mastered.  Many  teachers  make  the 
mistake  of  familiarizing  their  pupils  with  several  checks 


58  HOW  TO  TEACH  AEITHMETIC 

but  not  requiring  them  to  use  any  one  of  them  long  enough 
to  make  it  really  practical. 

Checks  for  Addition 

Checking  addition  by  combining  the  addends  in  the 
reverse  order  is  so  well  known  as  to  render  comment  un- 
necessary. To  combine  in  the  same  order  the  second  time 
as  the  first  is  practically  no  guarantee  of  the  correctness 
of  the  result.  The  mind  tends  to  repeat  its  own  mistakes, 
and  if  two  numbers  are  incorrectly  combined  the  first  time, 
the  chances  are  that  the  same  mistake  will  be  made  if  the 
order  of  the  addends  is  not  changed. 

Add  4823 
6792 
8437 
9265 
7426 
36743 

The  numbers  represented  by  the  first  column  may  be  added 
and  the  result  is  23.  The  sum  for  the  second  column,  with- 
out carrying,  is  22;  for  the  third  column  the  sum  is  25, 
and  for  the  fourth  it  is  34.  These  sums  may  be  represented 
in  either  of  the  following  manners  and  then  combined : 

23  34 

22  25 

25  22 

34  23 

36743  36743 

The  above  methods  are  frequently  used  by  persons  who  are 
likely  to  be  interrupted  while  adding. 


CHECKS  59 

Casting  Out  Nines 

9 

Another  simple  and  easily  remembered  check  for  addi- 
tion is  that  of  casting  out  the  nines.  The  following 
illustration  will  make  clear  the  method  of  applying  this 
check.  Suppose  we  wished  to  check  the  result,  36743,  in 
the  first  example  cited  above.  Add  the  digits  in  each  of  the 
addends  and  divide  the  sum  by  9.  The  remainder  is  called 
"the  excess. "  (It  is  the  remainder  which  would  be  found 
if  the  entire  number  were  divided  by  9.)  After  the  excess 
of  nines  in  each  addend  has  been  ascertained,  add  these 
excesses  and  divide  the  result  by  9.  The  remainder  (or 
excess)  thus  obtained  should  equal  the  excess  in  the  sum  in 
the  original  example. 

Excesses 

4823  8 

6792  6 

8437  4 

9265  4 

7426  _1 

36743  23  =  sum  of  the  excesses. 

Excess  in  the  sum  (36743)  equals  5.  This  is  also  the 
excess  in  the  sum  of  the  excesses. 

RULE. — The  excess  in  the  sum  must  equal  the  excess  in 
the  sum  of  the  excesses. 

Another  illustration  will  be  given  without  explanation : 

Excesses 

9426  3 

3875  5 

2873  2 

6941  2 

2739  3 


25854  15  =  sum  of  the  excesses, 


60  HOW  TO  TEACH  AEITHMETIC 

Excess  in  the  sum  (25854)  equals  6.    This  is  also  the  excess 
in  the  sum  of  the  excesses. 

After  the  check  has  been  applied  several  times,  the  pupil 
should  find  it  unnecessary  to  write  down  the  various  ex- 
cesses. After  a  brief  period  of  drill,  the  entire  check  can 
be  applied  without  writing  down  any  of  the  figures..  More- 
over, the  pupil  will  soon  discover  that  it  is  wise  to  group 
the  digits  of  a  number  into  combinations  whose  sum  is  9, 
and  that  the  grouping  enables  one  to  determine  the  excess 
in  a  number  very  quickly.  For  illustration,  it  is  desired 
to  find  the  excess  of  nines  in  32789.  The  practiced  eye  will 
group  the  7  and  the  2,  and  will  see  that  the  only  thing  to 
do  thereafter  is  to  determine  the  excess  of  nines  in  8  +  3. 
Instead  of  adding  the  digits  and  then  dividing  the  sum 
by  9,  pupils  should  accustom  themselves  to  add  the  digits 
until  the  sum  is  9,  or  more;  then  add  only  the  excess 
above  9. 

It  is  apparent  that  the  check  by  casting  out  the  nines  is 
not  a  proof  of  the  accuracy  of  the  result.  If  two  digits  in 
the  result  were  interchanged,  or  if  the  errors  were  of  such 
magnitudes  as  to  balance  each  other,  the  check  would  not 
detect  the  errors.  Moreover,  the  check  is  not  short,  and  it 
is  more  liable  to  error  than  a  new  addition,  unless  as  much 
time  is  given  to  drill  on  casting  out  of  nines  as  to  the 
addition  itself,  and  this  is  not  desirable. 

Checks  for  Subtraction 

Subtraction  may  be  checked  by  casting  out  the  nines,  but 
the  well-known  checks  of  adding  the  difference  to  the  sub- 
trahend to  produce  the  minuend,  or  of  subtracting  the 
difference  from  the  minuend  to  produce  the  subtrahend,  are 
more  easily  applied. 


CHECKS  61 

Checks  for  Multiplication 

Multiplication  may  be  checked  in  a  number  of  ways. 
Probably  the  methods  most  familiar  to  the  majority  of 
teachers  are  the  dividing  of  the  product  by  the  multipli- 
cand to  produce  the  multiplier,  or  dividing  the  product  by 
the  multiplier  to  produce  the  multiplicand.  The  check  by 
casting  out  the  nines  is  especially  adapted  to  multiplica- 
tion, and  is  easily  applied.  The  following  will  illustrate 
the  method  of  applying  this  check : 

Excess 

4836  3  (excess  of  nines  in  4836) 

285  6  (excess  of  nines  in    285) 


24180 
38688 
9672 
1378260  18  =  product  of  the  excesses. 

Excess  in  the  product  (1378260)  is  0,  which  is  also  the 
excess  in  product  of  the  excesses. 

RULE. — The  excess  of  nines  in  the  product  must  equal  the 
excess  in  the  product  of  the  excesses  of  the  multiplicand 
and  the  multiplier. 

A  second  illustration  follows : 


.    Excess 

3725      8  (excess  of  nines  in  3725) 
439      7  (excess  of  nines  in  439) 


33525 
11175 
14900 
1635275     56  =  product  of  the  excesses 


62  HOW  TO  TEACH  AEITHMETIC 

Excess  in  product  (1635275)  is  2,  which  is  also  the  excess 
in  the  product  of  the  excesses. 

Next  in  value  to  the  check  by  casting  out  the  nines  is 
that  of  interchanging  multiplicand  and  multiplier.  An 
illustration  will  make  this  clear. 

427  326 

326  427 


2562  2282 

854  652 

1281  1304 

139202  139202 

Checks  for  Division 

Division  may  be  checked  by  determining  whether  the 
result  obtained  by  multiplying  the  divisor  by  the  quotient 
and  adding  the  remainder,  if  any,  to  this  product,  equals 
the  dividend.  Casting  out  the  nines  may  be  used  to  advan- 
tage in  checking  division.  The  first  illustration  that  fol- 
lows shows  the  method  of  checking  division  when  t  there  is 
no  remainder.  The  second  shows  the  method  when  there 
is  a  remainder. 

Divide  1635275  by  439. 

439)1635275(3725 
1317 
3182 
3073 
1097 
878 
2195 
2195 

The  excess  in  the  dividend  (1635275)  is  2. 
The  excess  in  the  divisor  (439)  is  7. 
The  excess  in  the  quotient  (3725)  is  8. 


CHECKS  63 

The  product  of  the  excesses  in  the  divisor  and  quotient 
is  56,  and  the  excess  in  56  is  2.  This  is  also  the  excess  in 
the  dividend. 

RULE. — When  there  is  no  remainder,  the  excess  in  the 
dividend  equals  the  excess  in  the  product  of  the  excesses  of 
divisor  and  quotient. 

When  there  is  a  remainder,  the  check  is  applied  as 
follows : 

847)57426(67 
5082 
6606 
5929 
677 

Excess  in  the  dividend  is  6 ;  in  the  divisor  it  is  1 ;  in  the 
quotient  it  is  4 ;  and  in  the  remainder,  2. 

The  product  of  the  excesses  in  the  divisor  and  quotient 
(1x4)  is  4.  Add  the  excess  (2)  in  the  remainder;  the 
sum  is  6.  (Cast  the  nines  out  of  this  sum  if  it  is  greater 
than  9.)  The  result,  if  correct,  must  equal  the  excess  in 
the  dividend. 

When  one's  attention  is  first  directed  to  these  checks 
they  seem  quite  long  and  involved,  but  practice  will  enable 
one  to  use  them  with  facility. 

Approximate  Checks 

A  rough  or  approximate  estimate  of  the  result  before  a 
computation  is  made  is  frequently  a  valuable  check.  Too 
often  pupils  submit  an  answer  of  $480  when  the  result 
should  have  been  $4.80,  or  a  result  of  $54.16  when  the  cor- 
rect result  is  a  hundred  times  as  large.  If  teachers  would 
drill  their  pupils  more  frequently  in  approximating  results, 
the  number  of  absurd  and  impossible  answers  would  be 
reduced.  If  a  problem  requires  the  cost  of  520  yards  of 


64  HOW  TO  TEACH  AEITHMETIC 

cloth  at  13  cents  a  yard,  the  pupil  should  know  whether  the 
result  is  about  $6.50,  $65,  $650,  or  $6500.  If  the  problem 
requires  the  cost  of  432  articles  at  34  cents  each,  the  pupil 
should  see  that  since  the  cost  is  about  $£  each,  the  entire 
cost  will  be  about  one-third  of  $432,  or  $144. 

Self-Checking  Problems 

Self -checking  examples  and  problems  serve  a  useful  pur- 
pose, but  checking  should  not  be  confined  to  this  type.  The 
following  will  illustrate  this  type : 

3+  5+  7  +  6-21 
2+  8+  9+  3-22 
5+  4+  8+  9-26 
7+  2+  6+  5-20 


17  +  19  +  30  +  23-89 
The  admissions  to  a  certain  ball  park  were  as  follows : 

Wed.  Thurs.  Fri.  Sat.  Totals 

Children 428  369  514  483  1794 

Women 387  563  618  327  1895 

Men.... 942  1048  1127  1026  4143 

1757  1980  2259  1836  7832 

Numerous  self-checking  problems  based  upon  local  condi- 
tions will  suggest  themselves  to  most  teachers.  The  elec- 
tion returns,  the  enrollment  in  various  grades  of  the  school, 
the  number  of  pieces  of  mail  of  various  classes  handled  by 
the  local  postoffice  in  a  month,  are  among  the  problems  that 
may  be  used. 

Checking  Problems 

Eeasoning  problems  may  often  be  checked  by  solving  in 
two  ways  and  comparing  the  answers.  Instead  of  using 


CHECKS  65 

unitary  analysis,  proportion  may  be  used ;  instead  of  solv- 
ing a  problem  involving  cost  in  denominate  numbers  by 
the  use  of  aliquot  parts,  cancellation  may  be  used. 

Occasional  checking  is  of  only  slight  value.  The  pupil 
should  get  correct  results  and  should  know  that  the  results 
are  correct.  Habitual  checking  of  results  tends  to  beget 
confidence  in  one's  results,  and  confidence  in  one's  ability 
to  produce  correct  results  in  a  given  field  lies  near  the  basis 
of  success  in  that  field.  The  pupil  who  lacks  assurance  in 
himself  may  actually  retrograde  in  his  work  because  lack 
of  confidence  undermines  his  power  of  action. 


CHAPTER  V 
MARKING  PAPERS   IN  ARITHMETIC 

Objections  to  Marks 

No  system  for  marking  papers  has  been  devised  that  is 
free  from  objectionable  features.  Numerical  marks  are 
generally  used  to  represent  the  value  or  the  correctness 
of  the  result,  but  they  usually  put  no  value  upon  the  effort 
required  to  obtain  the  result.  Children  are  prone  to  com- 
pare one  grade  with  another  without  taking  into  considera- 
tion individual  differences  in  ability.  When  grades  are 
given  upon  the  quantity  of  work  done  in  a  given  time,  the 
gifted  child  will,  according  to  our  present  system,  receive 
a  higher  grade  than  the  dull  child,  but  the  grade  of  seventy 
or  eighty  that  the  dull  child  receives  may  represent  highly 
creditable  work  for  him. 

Teachers  generally  complain  because  of  the  amount  of 
labor  required  in  marking  papers.  This  labor  may  be 
unprofitable  and  even  harmful  to  the  teacher,  but  it  is  one 
of  the  necessary  burdens  of  successful  teaching.  The 
examination  of  papers  should  give  the  teacher  an  index  to 
the  attainment  of  the  pupils  and,  perhaps,  indirectly,  a 
measure  of  her  efficiency  in  teaching,  but  the  marks  and 
the  criticisms  given  should  serve  as  a  certain  stimulus  and 
incentive  to  the  pupils  to  work.  The  marks,  checks  and 
criticisms  given  should  not  be  looked  upon  as  mere  awards 
for  effort  or  achievement  but  as  suggestions  for  improve- 
ment. Wherever  they  do  not  serve  as  educational  stimuli, 
it  is  questionable  if  they  should  be  used. 

66 


MAEKING  PAPERS  IN  ARITHMETIC  67 

A  Daily  Memorandum 

Some  marking  is  necessary  for  mere  bookkeeping  pur- 
poses. With  large  classes  it  is  more  or  less  impossible  for 
the  teacher  to  remember  the  difficulties  and  needs  of  each 
individual  of  the  class.  Instead  of  assigning  grades  at 
the  close  of  the  recitation,  the  teacher  will  find  it  more 
practicable  and  helpful  if  she  keeps  a  memorandum  in  a 
book,  recording  the  entries  for  each  child  upon  a  separate 
page,  as  follows : 

CHARLES  GORDON 

Oct.  3.  Seems  uninterested,  but  understands  the  prin- 
ciple. 

Oct.  4.  Work  may  be  uninteresting  because  it  is  too  easy. 
Give  special  problems. 

Oct.  12.  Some  improvement.  Problems  may  be  too  ab- 
stract. Try  more  concrete  ones. 

Oct.  14.  Likes  to  make  his  own  problems,  but  doesn  't  know 
prices-  well  enough  to  make  his  problems  har- 
monize with  business  procedure. 

REX  SAFFER 

Oct.   5.     Lacks  facility  in  fundamentals. 

Oct.    7.     Assigned  problems  for  outside  practice. 

Oct.  8.  He  reported  the  work  done.  Gave  him  a  chance 
to  show  how  quickly  he  could  do  them.  Im- 
provement. Commended  him. 

Oct.   9.     Assigned  more  problems  for  outside  practice. 

Oct.  12.  Assignment  worked.  Takes  pride  in  it.  Continue 
the  practice. 

Such  records  as  these  would  be  of  great  value  to  the 
teacher  at  the  close  of  the  term  (in  determining  the  grades 
of  students). 


68  HOW  TO  TEACH  AEITHMETIC 

How  to  Get  the  Best  Results  from  Marks 

In  order  to  get  the  best  results  from  a  written  exercise, 
the  papers  should  always  be  marked  and  handed  back  to 
the  pupils.  Ordinarily  a  teacher  should  mark  but  not 
correct  the  mistakes.  That  is  the  pupil's  duty.  However, 
the  notes  of  the  teacher  should  be  clear  enough  to  enable 
the  pupils  to  correct  the  errors  intelligently.  It  is  well 
to  remember  that  calling  attention  to  a  fault  or  a  mistake 
once  does  not  mean  that  it  will  be  corrected.  Attention 
must  be  called  to  it  repeatedly  until  the  pupil  habitually 
gives  the  right  form.  A  clear  distinction  should  be  made 
between  errors  due  to  ignorance  and  those  due  to  careless- 
ness. Again,  not  all  the  faults  can  be  corrected  at  once. 
The  most  serious  mistakes  should  be  overcome  first;  the 
minor  ones  should  be  taken  up  later.  Perfection  is  attained 
by  gradual  growth. 

Stone's  Plan 

In  preparing  his  study  in  Arithmetical  Abilities,  Dr. 
Stone  used  the  original  scheme,  in  marking  the  papers,  of 
giving  a  score  for  each  step  in  a  problem.  This  is  an 
especially  good  scheme  to  employ  in  grading  the  funda- 
mentals in  the  primary  and  intermediate  grades.  Of  course 
such  a  system  of  grading  is  based  upon  the  assumption  that 
one  combination  is  of  equal  difficulty  with  another.  In 
such  a  plan  a  student  gets  credit  for  work  done  correctly. 
If  two  of  the  three  columns  are  added  correctly,  the  student 
gets  two  credits,  although  the  answer  as  a  whole  is  wrong. 
This  plan  could  not  be  applied  to  problems  unless  they  were 
of  equal  difficulty  or  were  graded  according  to  difficulty. 
At  present  we  have  no  scale  of  problems  for  measuring 
arithmetical  ability.  Such  a  scale  will  surely  be  forthcom- 
ing in  the  near  future. 


MARKING  PAPERS  IN  ARITHMETIC  69 

Lack  of  Uniformity  Among  Teachers  as  to  Marks 

The  authors  attempted  to  find  out  from  teachers  what 
the  current  practice  is  in  regard  to  the  marking  of  papers 
in  arithmetic,  but  they  found  no  common  mode  of  pro- 
cedure. Some  teachers  take  into  consideration  spelling 
and  punctuation ;  others,  the  minuteness  of  analysis ;  others, 
the  general  form  of  the  paper ;  and  still  others,  the  char- 
acter of  the  handwriting.  Although  the  solution  may  be 
mathematically  correct,  it  is  an  easy  matter  to  find  teachers 
who  refuse  to  give  full  credit  for  it,  if  there  is  a  misspelled 
word  in  the  solution  or  if  an  interrogation  point  is  omitted. 
At  times  this  matter  of  form  should  be  the  thing  of  para- 
mount importance,  but  to  place  great  emphasis  upon  it  in 
every  written  exercise  means  that  it  was  not  well  taught  at 
the  proper  time.  In  written  exercises  the  good  is  fre- 
quently the  enemy  of  the  best.  The  fact  that  the  first  copy 
should  be  good  enough  to  keep  should  be  impressed  upon 
the  pupils. 

Marking  the  Habit  and  Reasoning  Phases  of  Arithmetic 

In  the  habit  phases  of  arithmetic  there  are  only  two 
things  to  be  tested :  accuracy  and  rapidity.  If  the  various 
combinations  are  to  be  made  automatic,  rigid  tests  must  be 
made  repeatedly  to  avoid  wasteful  practices  and  to  insure 
progress.  It  is  quite  as  necessary  to  note  the  kind  and 
number  of  mistakes  made  as  to  note  the  number  of  com- 
binations correctly  made.  In  fact  the  number  of  mistakes 
and  the  frequency  with  which  given  combinations  are  erro- 
neously stated,  is  the  very  best  measure  of  accuracy.  If 
progress  is  being  made,  the  tests  given  by  the  teacher  will 
show  a  decrease  in  the  number  of  mechanical  errors. 

After  the  combinations  have  been  mastered,  the  pupils 
should  be  drilled  and  tested  for  rapidity.  The  best  results 


70  HOW  TO  TEACH  ARITHMETIC 

4 

will  usually  be  secured  if  the  pupils  compete  against  time. 
"With  watch  in  hand  the  teacher  directs  the  class  to  write 
certain  multiplication  tables  or  to  do  certain  sums  against 
time.  For  this  to  be  of  the  greatest  value  the  time  records 
of  each  individual  and  of  the  class  should  be  preserved  for 
future  comparisons.  The  amount  of  time  necessary  for  any 
given  type  of  work  should  decrease  rapidly  at  first  and 
more  gradually  as  the  pupils  approach  their  psychological 
limits. 

In  grading  problems  in  arithmetic  there  is  only  one  new 
element  to  be  added, — the  number  of  steps  in  the  problem. 
If  these  are  understood  in  their  sequence,  the  rest  of  the 
work  is  a  mere  matter  of  computation.  Until  we  have  a 
more  scientific  basis  for  estimating  these  steps,  they  may 
be  regarded  as  of  equal  value  or  of  equal  difficulty.  In 
grading  problems  which  require  interpretation  credit  may 
properly  be  given  both  for  interpretation  and  for  com- 
putation. 

It  is  usually  impossible  to  grade  work  of  any  kind  to 
within  one  percent.  The  marks  of  excellent,  very  good, 
fair,  not  satisfactory,  etc.,  usually  serve  the  purpose  better. 


CHAPTER  VI 
THE  NATURE  OF  PROBLEMS 

The  Changed  Character  of  Problems 

Text-book  makers  are  always  limited  in  the  selection  and 
organization  of  their  material  by  two  facts :  first,  changing 
social  conditions,  and,  second,  the  strengths  and  limita- 
tions of  the  different  maturity  levels  of  childhood.  The 
text-book  and  the  text-book  makers  are,  therefore,  inter- 
mediating agencies.  They  are  confronted  with  the  problem 
of  making  a  double  adjustment;  inwardly  to  the  human 
nature  of  childhood,  and  outwardly  to  those  social  forces 
and  conditions  that  largely  determine  the  nature  and 
organization  of  the  school. 

New  materials  come  into  the  curriculum  in  response  to 
new  needs,  or  in  response  to  old  needs  spreading  over 
wider  areas,  or  whenever  there  is  a  redistribution  of  the 
burdens  resting  upon  established  institutions.  Tradition 
and  reason  have  been  unable  to  set  a  fixed  division  of  work 
upon  social  institutions.  As  society  becomes  increasingly 
conscious  of  the  value  of  incidental  agencies  of  informal  edu- 
cation of  one  century,  it  frequently  gathers  them  up  and 
imposes  them  upon  the  schools.  For  this  reason  the  new 
arithmetics  contain  numerous  problems  concerning  the  sav- 
ing and  loaning  of  money,  mortgages,  improved  banking, 
building  and  loans  associations,  bonds,  rentals,  taxes,  public 
expenditures,  and  insurance.  Moreover,  there  is  an  increas- 
ing tendency  for  such  problems  to  be  arranged  and  pre- 
sented in  the  grammar  grades  to  correspond  to  the  occupa- 

71 


72  HOW  TO  TEACH  AEITHMETIC 

tions  they  illustrate.  Thus  we  have  a  list  of  problems  deal- 
ing with  agriculture,  another  with  banking,  another  with 
household  science,  and  another  with  trade  conditions.  This 
is  one  of  the  numerous  attempts  to  adjust  the  school  to 
vocational  life. 

It  is  also  held  that  material  thus  arranged  gives  the 
student  an  abundance  of  valuable  and  interesting  informa- 
tion, and  that  it  does  not  in  any  sense  diminish  his  oppor- 
tunities for  acquiring  arithmetical  skill.  Arithmetic  is  not 
taught  primarily  to  give  information  about  the  Keokuk 
Dam  or  the  Panama  Canal,  but  in  understanding  and 
comprehending  the  Keokuk  Dam  and  the  Panama  Canal 
some  arithmetic  is  necessary.  The  solution  of  occupational 
problems  does  not  rationalize  arithmetic;  such  problems 
merely  show  a  correlation  of  material  in  one  field  of  thought 
with  outside  events,  forces,  and  conditions.  Involved  in 
this  attempt  to  emphasize  the  social  and  industrial  phases 
of  arithmetic  are  two  highly  desirable  outcomes,  (1)  a 
knowledge  of  a  more  extended  application  and  use  of  arith- 
metical forms  and  skills  to  the  work  of  the  world,  and  (2) 
their  value  and  use  in  aiding  us  to  interpret  our  daily 
experiences. 

"Writers  of  arithmetics  make  little  effort  to  localize  ma- 
terials. This  is  clearly  the  province  of  the  teacher.  It  is 
easily  possible  for  teachers  to  carry  the  practice  to  an 
unnatural  extreme.  The  authors  knew  of  a  teacher  who 
lived  in  a  community  that  gave  considerable  attention  to 
poultry  raising.  Under  the  direction  of  this  teacher  the 
pupils  devoted  one  entire  term  to  the  preparation  of  orig- 
inal problems  dealing  with  poultry  raising.  Such  extrava- 
gant attempts  at  local  adjustment  as  this  are  regarded  as 
ridiculous  by  the  conservative  schoolmaster. 


THE  NATURE  OF  PROBLEMS  73 

Abstract  and  Concrete  Problems 

Problems  are  most  commonly  classified  as  abstract  or 
concrete.  There  is  a  tendency  to  increase  the  number  of 
concrete  problems.  In  the  lower  grades  these  problems  are 
selected  from  play,  school,  and  home  activities.  The  at- 
tempt is  made  to  introduce  no  problems  that  are  more 
difficult  or  complicated  than  those  the  children  use  in  their 
daily  life  outside  of  school.  In  this  there  is  a  well  pro- 
nounced danger — that  of  keeping  children  upon  a  low  level 
of  arithmetical  ability  until  outside  necessity  forces  them 
to  acquire  new  skills.  The  problems  used  in  school  should 
frequently  be  slightly  more  difficult  than  those  met  in  out- 
side life — should  require  some  "stretching  up."  Other- 
wise the  school  is  not  fulfilling  its  purpose  as  a  time  and 
labor  saving  device. 

On  the  other  hand,  some  have  an  erroneous  notion  as  to 
what  constitutes  $  concrete  problem.  Whenever  simple 
abstract  numbers  are  labeled  with  some  concrete  denomi- 
nation, like  dollars,  pounds,  or  acres,  it  is  assumed  that  the 
problem  has  been  made  concrete.  Such  an  assumption  may 
be  far  from  the  fact.  No  matter  how  concrete  the  labels 
may  be,  unless  the  facts  have  a  basis  in  the  child's  expe- 
rience, the  problem  is  abstract  to  him. 

There  is  still  a  third  way  in  which  the  tendency  to  make 
arithmetic  more  concrete  may  lead  to  harmful,  if  not  waste- 
ful, results.  It  is  the  practice  of  labeling  all  the  numbers 
in  the  processes  of  calculation.  To  illustrate : 

1.  Chester  had  13  marbles  and  his  father  bought  him  15 
more ;  how  many  had  he  then  ? 

2.  A  man  bought  7  head  of  cattle  at  $45  a  head ;  what 
did  they  cost  him  ? 

3.  If   $56    are    divided    equally    among    a    number    of 
boys,  so  that  each  boy  receives  $7,  how  many  boys  are 
there  ? 


74  HOW  TO  TEACH  ARITHMETIC 

4.  If  30  oranges  are  divided  equally  among  5  boys,  how 
many  will  each  get? 

(1)  (2)  (3)  (4) 

13  marbles     $  45  $7)jj^6  5  times)  30  oranges 

15  marbles  7  times  8  times  6  oranges 

28~marbles     $315 
• 
Referring  to  the  above  solutions,  Dr.  Faught  says :   '  '  The 

impression  seerns  to  prevail  that  the  processes  of  calcula- 
tion are  in  this  way  made  concrete  to  the  child.  These 
problems  can  be  solved  concretely ;  but  this  would  require 
us  to  count  the  marbles,  or  divide  the  oranges  among  the 
boys.  The  processes  of  calculation  are  intended  to  avoid 
this  method  of  solution ;  they  have  their  only  explanation 
in  the  fact  that  they  enable  us  to  obtain  certain  numerical 
results  more  economically  and  quickly  than  by  dealing  with 
the  actual  things  themselves.  But  is  there  no  place  for 
the  concrete  in  the  solution  of  problems?  To  be  sure,  the 
child  must  be  led  to  see  the  process  or  processes  implied 
in  the  problem  by  appealing  to  the  concrete,  to  objects, 
drawings,  etc.,  and  even  to  the  actual  things  themselves,  if 
necessary.  That  is  the  vital  point  in  the  solution  of  a 
problem,  and  the  one  at  which  so  many  teachers  fail.  But 
after  the  process  or  processes  have  been  determined,  it  is  a 
question  of  the  abstract,  of  symbols/7 

"The  word  marbles  does  not  help  the  child  in  thinking 
that  5  and  3  are  8,  or  the  $  in  thinking  that  8  sevens  are 
56.  Neither  does  he,  nor  should  he,  have  a  mental  image 
of  marbles  or  dollars  in  thinking  these  relations.  To  insist 
on  the  boy  counting  his  marbles  after  each  number  in  the 
first  solution  is  just  as  foolish  as  to  insist  on  the  boy 
counting  his  marbles  thus :  one  marble,  two  marbles,  three 
marbles,  four  marbles,  etc.  Boys  do  not  count  their  marbles 
that  way." 


THE  NATURE  OF  PROBLEMS  75 

The  method  described  in  this  quotation  is  the  one  chil- 
dren and  adults  use.  They  use  the  processes  of  calcula- 
tion in  the  shortest  possible  way  for  finding  numerical 
results.  They  never  stop  to  count  the  things ;  they  seldom 
know  or  try  to  remember  such  principles  as  "If  the  divi- 
dend and  divisor  are  both  concrete  the  quotient  is 
abstract, ' '  or  * '  The  product  is  always  of  the  same  denomi- 
nation as  the  multiplicand/'  Labels  certainly  do  not  assist 
in  calculations,  and  it  is  doubtful  if  they  aid  materially  in 
making  the  process  concrete.  It  is  no  more  difficult  to 
think  the  $45  as  abstract  in  the  multiplication  than  it  is 
to  think  of  cattle  as  abstract. 

Oral  and  Written  Problems 

Of  course  all  problems  are  mental  problems.  There  can 
be  no  other  kind.  "Mental  arithmetic "  is  a  misnomer. 
Usage  employs  it  to  mean  "oral  arithmetic."  Racially 
oral  arithmetic  must  have  preceded  written  arithmetic. 
Necessity  invented  the  abacus  to  take  the  place  of  counters, 
pebbles,  and  fingers.  After  writing  was  invented  and 
written  problems  were  introduced  a  long  struggle  began, 
yet  unsettled,  as  to  the  proper  distribution  between  the 
two  types  of  problems.  So  long  as  the  school  remains  sen- 
sitive to  social  changes  this  will  remain  an  unsolved  prob- 
lem. No  perfectly  satisfactory  adjustment  could  be  made 
unless  we  were  to  establish  a  monotonous  plane  of  life. 
Change  of  emphasis  is  inevitable.  The  number  of  oral 
problems  has  greatly  increased  in  recent  years,  perhaps 
relatively  more  rapidly  than  the  number  of  written  prob- 
lems. A  person  is  considered  illiterate,  or  practically  so, 
if  he  must  resort  to  paper  and  pencil  to  solve  the  simple 
problems  of  his  daily  life.  A  moment's  reflection  shows 
how  greatly  oral  problems  have  increased.  Every  store; 
the  grocery,  the  bakery,  the  hardware,  the  clothing,  and 


76  HOW  TO  TEACH  AEITHMETIC 

general  merchandise  stores,  sell  numerous  articles,  for- 
merly considered  as  luxuries,  that  are  now  commonly  re- 
garded as  necessities.  Consequently,  the  average  person 
is  compelled  to  solve  a  greater  variety  of  oral  problems 
than  formerly.  No  doubt  as  society  grows  more  complex 
and  varied  in  nature  the  number  of  such  problems  will 
continue  to  increase.  Writers  of  text-books  in  recognition 
of  this  new  demand  now  give  long  lists  of  oral  problems 
drawn  from  daily  life.  In  one  of  the  newer  sets  of  arith- 
metics 1800  such  problems  are  given. 

In  this  connection  it  is  proper  to  correct  the  impression 
of  those  teachers  who  think  that  rapid  work  on  written 
problems  implies  a  corresponding  degree  of  speed  and 
accuracy  in  oral  problems.  Reliable  investigations  show 
conclusively  that  a  person  may  be  rapid  in  written  work 
and  slow  in  oral  work.  It  follows  as  a  matter  of  course 
that  many  oral  problems  are  necessary  if  facility  and  skill 
are  expected  in  the  purely  mental  operations. 

We  have  alluded  to  the  form  of  solution  of  written  prob- 
lems, and  have  urged  teachers  to  avoid  the  superfluous 
exercises  not  practiced  by  business  men  in  getting  the 
"answer."  We  should  be  seriously  at  fault  and  open  to 
criticism  if  we  failed  to  urge  the  necessity  of  rationalizing 
and  explaining  most  of  the  mechanical  operations  and  of 
avoiding  inaccuracies  of  statement.  Analysis  by  steps  is 
needed  to  familiarize  children  with  the  reasoning  proc- 
esses involved  in  the  solution  of  the  various  types  of  prob- 
lems. The  virtue  of  an  analysis  is  not  in  its  length,  but  in 
its  directness.  Direct  analysis  means  a  greater  reliance 
on  oral  work. 

Solution  of  Problems 

It  is  frequently  good  practice  to  separate  the  solution 
from  the  operation.  Pupils  may  indicate  the  separate  steps 


THE  NATURE  OF  PROBLEMS  77 

they  would  take  in  solving  the  problem  before  they  attempt 
to  do  the  work.  Then  the  solution  may  be  stated  free  of 
labels,  for  example: 

1.  William  had  7  marbles;  his  sister  gave  him  4  more, 
and  he  traded  3  off  for  a  top ;  he  then  found  14  and  lost 
half  of  what  he  then  had?     How  many  had  he  left? 

SOLUTION  : 

7  +  4  23  +  14  =  11 

11  equal  the  number  of  marbles  he  had  left. 

2.  A  field  is  80  rods  long  by  150  rods  wide.    How  many 
dollars  is  it  worth  at  $20  an  acre? 


SOLUTION  : 

80x150x20 


160 


-1500, 


which  is  the  number  of  dollars  this  land  is  worth. 

The  practice  of  having  pupils  indicate  the  operations 
should  be  continued  in  the  upper  grades.  Whenever  the 
mode  of  procedure  is  clear  and  the  habit  is  fixed  for  a 
given  type  of  problem,  the  practice  may  be  discontinued. 
Such  a  device  discourages  the  " trial  and  error"  method 
normally  used  by  children. 

Examples  and  Problems 

The  word  "problem*?  is  used  in  some  sections  of  the 
country  to  mean  a  statement  from  which  the  pupil  must 
first  decide  what  operations  are  to  be  performed  and  what 
their  sequence  should  be  before  proceeding  to  the  calcula- 
tions. In  an  example  the  mathematical  symbols  tell  the 
pupil  what  to  do,  whether  to  add,  subtract,  multiply,  or 
divide. 


78  HOW  TO  TEACH  AEITHMETIC 

"An  example  is  a  prereasoned  problem."  Chronologi- 
cally problems  must  have  preceded  examples.  The  race 
was  confronted  with  a  multitude  of  interesting  problems 
long  before  it  invented  a  mathematical  symbolism  to  express 
number  relations.  Examples  thus  became  devices  for  fix- 
ing the  identical  and  constantly  recurring  elements  in 
problems.  In  spite  of  this  obvious  relationship,  they  are 
frequently  sharply  separated  and  taught  as  if  neither  had 
much  dependence  upon  the  other. 

Logically,  problems  should  both  precede  and  follow 
examples  in  instruction.  The  formal  phases  may  be 
abstracted  for  memorization,  when  pupils  are  reasonably 
familiar  with  them  in  their  problematic  setting,  but  they 
should  be  restored  to  problems  later  to  insure  their 
application. 

Examples  are  abstract ;  problems  are  concrete.  Examples 
are  expressed  by  signs ;  problems  by  descriptive  statements. 
In  a  problem  the  pupil  has  the  double  duty  of  reasoning 
and  manipulating  in  either  of  which  he  may  err.  Examples 
ar*e  thus  mechanical,  problems  are  rational.  An  example 
may  be  an  illustration  of  some  problem  or  principle  that 
has  been  demonstrated;  a  problem  requires  solution. 
Examples  help  to  fix  things ;  problems,  to  insure  growth. 

Problems  Without  Numbers 

Problems  without  numbers  develop  clear  imaging  and 
accurate  reasoning.  The  pupil  must  concentrate  upon  the 
reasoning  processes,  as  there  are  no  numbers  to  confuse  or 
mislead.  Such  problems  tend  to  develop  the  ability  to 
decide  what  operation  to  use  under  given  conditions.  There 
is  a  growing  demand  for  more  problems  of  this  kind.  Any 
resourceful  teacher  can  manufacture  an  abundance  of  them. 
We  give  a  few  illustrative  examples: 


THE  NATURE  OF  PROBLEMS  79 

If  you  know  the  height  of  a  flagstaff  on  which  there  is 
a  mark,  and  how  far  it  is  from  the  ground  to  this  mark, 
how  can  you  find  how  far  it  is  from  the  mark  to  the  top 
of  the  flagstaff? 

To  find  the  area  of  a  square,  what  must  you  know  and 
what  must  you  do? 

If  you  know  the  dimensions  of  a  square,  how  can  you 
find  its  perimeter? 

If  you  know  the  area  of  a  triangle  and  the  length  of  the 
base,  what  else  must  you  know  and  what  must  you  do  to 
find  the  altitude  of  the  triangle  ? 

If  you  know  the  perimeter  and  two  sides  of  any  triangle, 
how  can  you  find  the  remaining  side? 

I  know  how  many  feet  long  and  wide  a  cellar  is  to  be, 
what  else  must  I  know  and  what  must  I  do  to  find  how 
many  wagon  loads  of  earth  will  be  taken  out  in  making  the 
excavation  ? 

I  have  some  money,  I  know  the  cost  of  one  hat ;  how  am 
I  to  find  how  many  hats  I  can  buy  at  the  same  rate  with 
the  whole  of  my  money? 

B  owns  a  triangular  lot ;  he  knows  the  length,  in  rods, 
of  its  base,  altitude,  and  hypotenuse.  How  shall  he  find 
how  many  acres  it  contains? 

John  weighed  a  basket  of  wood ;  after  it  stood  out  in  the 
rain  over  night  he  weighed  it  again.  How  can  he  find  the 
weight  of  water  absorbed? 

H  holds  a  certain  number  of  books  in  his  hand  and  tells 
W  that  they  are  one-half  per  cent  of  all  his  books.  How 
can  W  find  how  many  books  H  has? 

A  and  B  are  a  certain  number  of  miles  apart  and  are 
traveling  in  the  same  direction.  We  know  the  number  of 
miles  per  hour  that  each  travels  and  that  B  travels  the 
faster.  How  shall  we  find  the  number  of  hours  required 
for  B  to  overtake  A? 


80.  HOW  TO  TEACH  ARITHMETIC 

I  know  the  length  of  a  line  of  fence,  and  the  number  of 
posts,  counting  the  end  ones.  How  shall  I  find  the  dis- 
tance from  one  post  to  the  next  ? 

What  must  you  know  and  what  must  you  do  to  find  how 
many  times  a  wagon  wheel  will  turn  in  going  two  miles  ? 

If  you  know  the  length  of  the  edge  of  a  cube,  what  can 
you  find  and  how  would  you  do  it? 

A  cow  is  tied  to  the  corner  post  of  a  square  lot.  Given 
the  length  x>f  the  rope,  how  can  you  find  the  area  of  ground 
over  which  she  can  graze  if  she  is  inside  the  lot?  If  she 
is  outside  the  lot? 

If  you  know  two-thirds  of  a  certain  number,  how  can 
you  find  the  number? 

If  you  know  how  many  quarts  of  milk  a  family  uses  each 
day  and  the  price  per  quart,  how  can  you  find  the  amount 
of  the  milk  bill  for  the  month  of  January? 

If  you  know  the  cost  of  a  bushel  of  anything  and  wish 
to  know  the  cost  of  a  peck,  at  the  same  rate,  how  do  you 
proceed  ? 

What  must  you  know  and  what  must  you  do  to  find 
how  many  bushels  of  wheat  a  bin  will  hold? 

If  you  know  the  depth  of  rain-fall  on  a  given  field  for 
a  given  time,  what  else  must  you  know  and  what  must  you 
do  to  find  the  weight  of  the  water?  ' 

What  must  you  know  and  what  must  you  do  to  find  out 
how  many  bunches  of  shingles  would  be  used  in  putting 
a  new  roof  on  your  house  ? 

If  you  know  how  much  a  man's  average  expenses  are  for 
a  month  what  must  you  know  and  what  must  you  do  to 
find  out  how  much  he  saves  in  one  year  ? 

What  must  you  know  and  what  must  you  do  to  find  the 
balance  in  Mr.  A's  bank  account  at  the  end  of  a  given 
month  ? 

If  you  know  the  cost  of  a  ticket  from  your  home  to  New 


THE  NATURE  OF  PROBLEMS  81 

York  City  what  else  must  you  know  and  what  must  you 
do  to  find  the  average  cost  of  the  trip  per  mile? 

What  must  you  know  and  what  must  you  do  to  find  the, 
cost  of  laying  a  cement  sidewalk  in  front  of  your  home? 

What  facts  must  be  known  and  how  can  they  be  used 
in  order  to  compute  the  average  attendance  at  your  school 
for  the  year? 

Such  work  as  has  been  suggested  above  gives  the  pupils 
a  better  appreciation  of  what  the  various  arithmetical 
processes  do  than  they  otherwise  have.  It  emphasizes  the 
importance  of  knowing  exactly  what  is  to  be  done  before 
attempting  to  do  it.  It  makes  its  first  appeal  to  the  under- 
standing rather  than  to  the  memory. 

Original  Problems 

Children  should  be  permitted  to  "make  up  problems." 
This  is  another  excellent  device  for  stimulating  vigorous 
thinking.  The  best  of  the  original  problems  might  be 
written  in  a  book  called  "Our  Original  Arithmetic"  or 
"Our  Own  Arithmetic."  The  problems  thus  preserved 
may  be  used  later  for  review  purposes.  Whenever  the 
data  involved  in  such  problems  are  of  an  informational 
character  they  should  correspond  to  actual  conditions. 
For  example  it  is  not  wise  to  assume  in  a  problem  that 
the  distance  from  New  York  to  Chicago  is  250  miles ;  that 
wheat  sells  for  $6  a  bushel  and  silk  for  $85  a  yard.  The 
duplication  of  a  difficult  problem  by  a  simpler  original 
problem  frequently  clears  away  the  difficulty. 

NOTE — For  good  selections  of  problems  similar  to  the  above  the 
reader  is  referred  to  "Problems  Without  Figures, "  by  S.  Y.  Gillan; 
published  by  S.  Y.  Gillan  and  Company,  Milwaukee,  Wis. 


CHAPTER  VII 

RULES  AND  ANALYSES 

Not  many  decades  ago  a  teacher  who  could  perform  long 
and  involved  computations  and  could  set  them  down  in 
neat  form  was  considered  a  master  of  arithmetic.  The  rote 
system  of  learning  was  in  vogue  and  more  emphasis  was 
placed  upon  forms  and  symbols  than  upon  the  things 
symbolized.  Herbert  Spencer  said  that  to  repeat  the 
words  correctly  was  everything,  to  understand  the  mean- 
ing was  nothing,  and  thus  the  spirit  was  sacrificed  to  the 
letter.  Text -books  in  arithmetic  contained  numerous  defi- 
nitions and  rules  which  the  pupils  were  expected  first  to 
memorize  and  then  apply.  As  an  aid  to  the  memory  many 
definitions  and  rules  were  put  into  rhyme.  Rhyming  arith- 
metics became  very  common  during  the  seventeenth  cen- 
tury. The  following  from  the  "Handmaid  of  Arithmetic 
Refined,"  published  by  Nicholas  Hunt  in  1633,  will 
illustrate : 

"We  are  taught  in  numeration,  number  riting  and  notation. 

Add  thou  upright,  reserving  every  ten, 
And  rite  the  digits  down  all  with  the  pen. 

Subtract  the  lesser  from  the  great,  noting  the  rest, 
Or  ten  to  borrow  you  are  ever  prest, 
To  pay  what  borrowed  was  think  it  no  paine, 
But  honesty  redounding  to  you  gaine. " 

It  is  not  necessary,  however,  to  go  farther  back  than  the 
nineteenth  century  to  find  examples  of  these  rhyming 
arithmetics.  A  book  compiled  by  John  Graham  (an  Ameri- 

82 


RULES  AND  ANALYSES 


83 


can)  in  1824,  entitled,  "The  Farmer's  and  Mechanic's 
Assistant  and  Companion;  or,  a  New  System  of  Decimal 
Arithmetic,  adapted  for  the  easy  and  regular  instruction 
of  the  youth  in  the  United  States,"  was  written  partly 
in  verse.  The  author  says :  "I  have  endeavored,  for  the 
encouragement  of  the  learner,  to  do  all  that  I  possibly 
could;  the  rules  are  all  very  plain  and f easy  to  be  under- 
stood to  which  I  would  advise  every  scholar  particularly 
to  attend;  for  I,  myself,  have  observed, 

"  Study  the  rule,  your  question  pry, 
You'll  gain  the  answer  by  and  by. *J 

A  little  further  on  he  says: 

"This  little  book,  peruse  it  well;  I  hope  in  it  you'll  find 
Something  entertaining,  useful  work,  to  cultivate  the  mind; 
From  it  you  may,  if  well  applied,  some  information  gain; 
Arithmetical  rules  you'll  find,  both  easy,  short  and  plain, 
My  best  advice  to  youth  I  give  improve  your  golden  span; 
Seek  for  knowledge  while  you  're  young — education  makes  the  man. ' ' 

Here  follows  one  of  the  most  interesting  problems. 


'Dear  friend,  I  request  you  with  caution  and  care, 
To  measure  this  lion  exact  to  the  hair, 
His  head  is  twelve  inches,  from  his  ears  to  his  nose; 
This  measure  I  give  you  the  rest  to  disclose. 


84  HOW  TO  TEACH  AEITHMETIC 

His  tail  is  as  long  as  his  head,  and  a  half 

Of  his  body  is  the  length  of  his  head  and  his  tail; 

He's  a  surly  old  rogue,  yet  you  can  not  well  fail 

To  tell  his  whole  length,  when  his  substance  you  see; 

You  've  the,  length  of  his  head,  as  was  given  to  me. 

The  question  required,  is  separately  to  tell 

His  whole  length,  his  body's  length,  and  the  length  of  his  tail." 

Rules  were  too  often,  both  the  beginning  and  the  end 
of  teaching.1  It  is  true  that  efficiency  in  a  subject  like 
arithmetic  presupposes  that  many  processes  can  be  used 
mechanically,  but  it  should  not  be  inferred  that  our  knowl- 
edge of  all  processes  in  arithmetic  should  be  mechanical. 

"We  should  both  educate  and  train.  DeMorgan  said. 
"The  merely  showing  the  student  a  rule  by  which  he  is 
to  work,  and  comparing  his  answer  with  a  key  to  the  book 
is  not  teaching  arithmetic  any  more  than  presenting  him 
with  a  grammar  and  a  dictionary  is  teaching  him  Latin. ' ' 2 

Rationalizing  the  Processes 

One  of  the  tendencies  in  the  teaching  of  arithmetic  to- 
day is  the  attempt  to  rationalize  the  various  processes. 
Every  rule  in  arithmetic  rests  upon  some  principle  and  a 
topic  should  be  introduced,  in  the  later  grammar  grades 
especially,  in  such  a  way  as  to  reveal  this  principle.  It  is 
unquestionably  true  that  the  rationalization  of  some  proc- 
esses is  difficult  for  pupils  at  the  age  when  these  proc- 
esses should  be  learned.  A  process  may  properly  be  ra- 
tionalized whenever  a  pupil  of  ordinary  ability,  by  whom 
it  is  to  be  mastered,  can  comprehend  it.  Some  pupils  will 
profit  but  little  by  this  attempt  at  rationalization,  but  this 
does  not  justify  the  teacher  in  omitting  the  explanation  and 

1  Spencer  said  that  he   doubted  if  one  boy   in  five  hundred  ever 
heard  the  explanation  of  a  rule  in  arithmetic. 

2  DeMorgan,  ' '  Studies  in  Mathematics, ' '  p.  22. 


RULES  AND  ANALYSES  85 

presenting  the  subject  in  a  mechanical  way.  Many  chil- 
dren can  early  be  trained  to  look  for  reasons  and  laws. 
"The  pupil  is  expected  in  a  sense  to  rediscover  the  sub- 
ject, though  not  without  profit  from  the  fact  that  the  race 
has  already  discovered  it.  The  pupil  is  a  child  tottering 
across  the  room,  not  a  Stanley  penetrating  into  the  heart 
of  Africa.  The  teacher  stands  before  him  and  with  word 
and  smile  entices  him  on ;  selecting  his  path,  choosing  every 
spot  where  he  is  to  plant  his  foot,  catching  him  when  he 
stumbles,  raising  him  when  he  falls,  but  when  he  has 
crossed  the  room  he  has  done  it  himself  and  has  made 
more  progress  towards  walking  whither  he  would  than  if 
he  had  been  carried  across  the  room,  or  across  hundreds 
of  rooms,  or  even  into  the  heart  of  Africa."  1 

Telling  is  frequently  not  teaching.  The  phrase  i '  go  apd 
see"  has  been  revised  to  read  "see  and  think." 

No  one  doubts  that  children  can  think  and  do  think  about 
matters  that  are  within  their  capacities.  All  normal  pupils 
have  a  natural  desire  for  knowledge  and  this  desire  may 
be  increased  and  intensified.  It  is  one  of  the  duties  of  the 
teacher  to  make  the  pupil  conscious  of  his  powers;  this 
cannot  be  accomplished  by  continually  feeding  him  on  an 
easy  intellectual  diet.  Colonel  Parker  said  that  many  nat- 
urally capable  children  are  "helped  into  helplessness."  The 
teacher  who  merely  stores  the  pupil's  mind  with  facts  has 
done  only  a  part  of  his  duty.  The  pupil  should  be  so 
trained  that  he  will  have  the  desire  to  acquire  more  knowl- 
edge for  himself  and  to  use  this  knowledge  in  an  efficient 
manner.  The  teacher  should  seek  to  make  the  pupil  in 
the  later  grammar  grades  dissatisfied  with  knowing  only 
the  what  and  the  how,  and  should  emphasize  the  why. 

We  recognize  to-day  that  the  thought  side  of  arithmetic 
should  receive  greater  emphasis  than  it  did  a  few  genera- 

i"The  Teaching  of  Mathematics, "  pp.  69-70.  J.  W.  A.  Young. 


86  HOW  TO  TEACH  ARITHMETIC 

tions  ago.  We  are  breaking  away  from  the  rules  of  the 
subject  and  are  directing  more  attention  to  the  principles 
that  underlie  the  rules.  Arbitrary  rules  committed  to 
memory  cannot  develop  a  thinker. 

It  is  much  easier  to  say  to  the  pupils, — learn  this  rule 
and  work  this  list  of  problems  by  it, — than  it  is  to  teach 
him  to  think  independently.  The  indolent  teacher  will 
usually  choose  the  path  of  least  resistance  and  will  teach 
arithmetic  in  a  mechanical  way. 

Too  many  pupils  in  arithmetic  are  short-circuited  and 
taught  results  only.  In  many  schools  pupils  are  allowed 
to  combine  the  numbers  in  such  a  way  as  to  get  the  result 
and  nothing  more  is  expected  or  required.  No  one  ques- 
tions the  desirability  of  getting  correct  results,  but  when 
"getting  the  result"  is  considered  as  of  prime  importance 
and  the  principles  underlying  the  processes  are  regarded 
as  of  little  importance  we  tend  to  develop  the  art  of 
ciphering  at  the  expense  of  the  reasoning  faculty.  It  is 
not  enough  simply  to  do  in  mathematics.  It  is  important 
to  know  also  when  a  process  should  b'e  used  and  to  under- 
stand the  reasoning  upon  which  the  process  is  based. 

If  the  principles  underlying  the  various  processes  are 
understood  and  a  fair  mastery  of  the  fundamental  opera- 
tions is  acquired  in  the  first  four  or  five  grades,  the  result 
will  in  many  cases  take  care  of  itself.  When  the  higher 
purposes  are  served  the  lower  ones  are  usually  taken  care 
of  also.  Instead  of  requiring  pupils  in  the  later  gram- 
mar grades  to  learn  numerous  arbitrary  rules  let  us  explain 
the  various  processes  to  them,  showing  how  each  step  is 
based  on  something  that  precedes  and  how  it  is  to  be 
used  in  that  which  follows.  These  are  matters  that  are 
worth  while.  Any  method  of  teaching  that  thwarts  the 
natural  movement  of  the  mind  is  not  efficient.  Under- 
standing should  usually  come  before  practice.  The  rever- 


RULES  AND  ANALYSES  87 

sal  of  this  order  seldom  leads  to  accumulation  of  facts  or 
the  development  of  power. 

Place  of  Rules  in  Arithmetic 

Rules  have  a  very  important  place  in  the  modern  teach- 
ing of  arithmetic.  A  rule  should  come  at  the  close  and 
not  at  the  beginning  of  a  process.  The  rule  should  be 
reached  and  formulated  by  the  pupil  himself  under  the 
direction  of  the  teacher.  It  should  be  a  brief  and  concise 
statement  of  procedure  for  future  guidance.  The  pupil 
who  appreciates  the  full  significance  of  the  favorite 
aphorism  of  Lucretia  Mott — ' '  Truth  for  authority  and  not 
authority  for  truth,"  as  it  applies  to  arithmetic,  has  the 
proper  insight  into  the  subject. 

The  pupil  who  has  been  taught  his  arithmetic  by  mem- 
orizing dogmatically  stated  rules  sees  but  little  unity  in  the 
subject.  He  is  prone  to  depend  upon  typical  cases  and  to 
associate  a  process  with  certain  phraseology.  The  pupil 
who  has  learned  his  arithmetic  by  mastering  the  principles 
of  the  subject  is  not  in  serious  difficulty  every  time  he 
encounters  a  new  type  of  problem,  and  an  insignificant 
change  in  phraseology  does  not  mislead  him.  Any  knowl- 
edge which  the  pupil  has  acquired  himself  becomes  more 
thoroughly  his  own  than  it  could  otherwise  be.  "  While 
rules  lying  isolated  in  the  mind — not  joined  to  its  other 
contents  as  outgrowths  from  them — are  continually  for- 
gotten, the  principles  which  these  rules  express  piecemeal 
become,  when  once  reached  by  the  understanding,  endur- 
ing possessions.  "While  the  rule-taught  youth  is  at  sea 
when  beyond  his  rules,  the  youth  instructed  in  principles 
solves  a  new  case  as  readily  as  an  old  one.  Between  a  mind 
of  rules  and  a  mind  of  principles,  there  exists  a  difference 
such  as  that  between  a  confused  heap  of  materials  and  the 


88  HOW  TO  TEACH  ARITHMETIC 

same  materials  organized  into  a  complete  whole  with  all 
its  parts  bound  together. ' ' x 

Analysis 

If  it  is  desirable  to  emphasize  the  reasons  underlying 
the  various  processes,  making  each  step  rational  to  the 
pupil,  it  is  necessary  that  some  power  of  analysis  should 
be  developed.  The  power  to  analyze  and  to  relate  is  one 
of  the  essentials  of  the  clear  thinker.  The  difficulty  in  a 
process  or  in  a  problem  usually  lies  in  the  fact  that  the 
pupil  lacks  the  ability  to  break  it  up  into  its  several  parts 
and  to  apprehend  the  relation  between  the  parts.  Unless 
a  pupil  has  the  power  to  analyze,  his  knowledge  of  arith- 
metic will  be  more  or  less  mechanical.  He  will  depend 
upon  type  forms  and  similar  cases  and  when  these  cannot 
be  found  he  is  involved  in  difficulty.  A  pupil  who  has 
the  power  to  analyze  is  much  less  dependent  upon  rules 
and  type  forms.  He  is  frequently  able  to  devise  a  solu- 
tion of  his  own  and  he  relies  less  upon  teacher  and  text. 
The  ability  to  discover  relations  and  to  make  proper  com- 
parisons gives  one  relative  freedom  from  mechanical  and 
initiative  procedures. 

/ 

Essentials  of  an  Analysis 

An  analysis  is  an  orderly  statement  of  the  facts;  if  the 
facts  are  understood  and  their  sequence  is  apprehended 
the  analysis  can  be  stated.  Every  step  in  a  good  analysis 
is  a  judgment  and  is  related  directly  or  indirectly  to  every 
other  step.  An  analysis  should  be  so  concise  that  if  a  step 
is  omitted  no  further  progress  can  be  made.  Analysis  in 
arithmetic  may  be  over-emphasized,  but  this  should  not 

i  Herbert  Spencer,  ' '  Education, »  p.  103. 


EULES  AND  ANALYSES  89 

condemn  analysis;  it  should  argue  for  more  wisdom  in  its* 
use.  Excess  is  always  to  be  avoided.  No  thoughtful  teacher 
would  require  a  pupil  to  analyze  every  problem.  To  do 
this  would  be  almost  as  great  a  mistake  as  to  require  no 
analysis.  The  correct  solution  of  a  problem  is  usually 
evidence  that  a  proper  analysis  has  been  made.  It  is  pos- 
sible to  train  pupils  to  analyze  minutely  and  yet  these 
same  pupils  may  have  but  little  grasp  of  number  rela- 
tions. It  is  not  desirable  to  emphasize  either  mechanical 
computation  or  analysis  at  the  expense  of  the  other.  Both 
are  very  important  and  should  receive  due  consideration. 
Pupils  should  frequently  be  required  to  solve  problems 
by  the  shortest  possible  method  with  no  formal  analysis 
or  explanation.  At  such  times  rapidity  and  accuracy  of 
solution  should  be  regarded  as  of  paramount  importance. 

The  pupil  who  has  the  power  to  analyze  minutely  but 
who  performs  the  mechanical  operations  slowly  and  inac- 
curately is  as  truly  to  be  pitied  as  the  one  who  performs 
these  operations  with  speed  and  accuracy  but  has  little 
ability  in  seeing  relationships. 

Analyses  are  sometimes  necessary  to  reveal  the  line  of 
the  pupil's  reasoning  to  the  teacher.  It  is  well  at  times 
to  have  the  pupils'  mental  processes  exposed.  Such  a  pro- 
cedure frequently  tends  to  clarify  difficulties  in  the  mind 
of  the  pupil.  Many  problems  which  seem  very  difficult 
and  involved  are  easily  solved  when  a  detailed,  analysis  is 
attempted. 

Type  Forms  of  Analysis 

Some  teachers  believe  that  a  certain  type  form  of  analysis 
should  always  be  used  in  the  solution  of  a  given  type  of 
problem.  They  insist  that  the  pupil  shall  always  use  the 
exact  words  of  a  model  analysis  which  has  been  explained. 
It  is  a  mistake  to  insist  upon  a  rigid  and  inflexible  form 


90  HOW  TO  TEACH  AEITHMETIC 

in  the  solution  of  any  problem.  There  is  no  type  form 
that  is  best  for  all  pupils  or  for  all  problems.  Too  great 
insistence  upon  particular  ways  of  doing  things  tends  to 
check  originality  and  initiative.  An  analysis  should  be 
considered  as  a  means  to  an  end  not  an  end  in  itself.  It 
is  not  unwise  to  direct  the  attention  of  the  pupils  occa- 
sionally to  certain  type  analyses.  Such  models  may  serve 
as  a  good  basis  for  others,  but  there  should  be  no  insis- 
tence upon  the  adoption  of  a  particular  phraseology. 
Whatever  form  most  concisely  expresses  the  ideas  in- 
volved should  be  used  and  flexibility  of  thought  and  of  ex- 
pression should  be  encouraged. 

Too  many  words  often  betray  a  poverty  of  mind  or  lack 
of  a  clear  comprehension  of  the  data  involved.  A  multi- 
plicity of  words  may  obscure  the  thought.  "It  is  of  little 
importance  how  the  pupil  begins  or  how  he  ends  the 
analysis,  or  whether  he  puts  in  the  requisite  number  of 
'sinces'  and  'theref ores'  if  only  he  has  been, — direct,  clear, 
concise,  coherent,  and  grammatical. ' ' x 

Unitary  Analysis 

The  unitary  analysis  is  of  value,  but  to  require  the  pupil 
to  use  it  in  the  solution  of  every  problem  is  a  great  mis- 
take. If  a  pupil  really  sees  the  relations  between  the  mag- 
nitudes involved  he  will  not  often  need  the  unitary  anal- 
ysis. Such  an  analysis  is  seldom  simpler  than  a  solution 
by  means  of  a  direct  comparison.  If  a  pupil  is  required  to 
solve  a  problem  like  the  following:  "If  14  spools  of 
thread  cost  70  cents,  what  will  42  spools  of  thread  cost 
at  the  same  rate?"  he  should  see  at  once  that  forty-two 
spools  will  cost  three  times  as  much  as  fourteen  spools, 
and  he  should  be  encouraged  to  use  this  short  cut 

i  Longan  's  ' '  First  Lessons  in  Arithmetic, ' '  p.  8. 


RULES  AND  ANALYSES  91 

to  secure  the  result.  In  solving  such  a  problem  there  is 
a  loss  of  time  in  first  finding  the  cost  of  one  spool. .  If  the 
problem  stated  the  cost  of  14  spools  and  required  the  cost 
of  33  spools  at  the  same,  rate,  no  time  would  be  lost  by 
first  finding  the  cost  of  one  spool.  If  the  cost  of  twenty 
articles  is  given  and  the  pupil  is  required  to  find  the  cost 
of  ten  articles  at  the  same  rate  he  should  see  at  once  that 
the  required  cost  is  one-half  of  the  stated  cost. 

Mechanical  Analysis 

Most  teachers  have  heard  of  the  pupil  who  when  called 
upon  to  give  an  analysis  for  a  certain  problem  arose  but 
was  silent.  The  teacher  said,  " Don't  you  know  how  to 
analyze  that  problem?  It  is  the  kind  that  begins  with 
'since.'  '  With  this  cue  the  pupil  at  once  went  through  a 
so-called  analysis  and  the  teacher  commended  him  for  his 
work.  Such  a  procedure  should  not  be  dignified  by  the 
name  of  "  analysis. "  The  pupil  was  only  repeating  a 
memorized  type  form.  The  analysis  had  little  or  no  con- 
tent in  his  mind.  He  didn't  understand  the  gist  of  it. 
An  advertisement  frequently  seen  to-day  contains  a  state- 
ment that  is  in  point  in  this  connection,  "You  may  teach 
a  parrot  to  say  'just  as  good/  but  he  won't  know  what  he 
is  talking  about."  Many  analyses  which  seem  to  involve 
a  good  deal  of  thought  are  repeated  mechanically  when  the 
proper  cue  is  given. 

Encourage  the  pupil  to  seek  for  the  briefest  and  best 
form  of  analysis.  By  the  use  of  judicious  questions  make 
sure  that  the  analysis  has  the  proper  content  in  his  mind. 
Encourage  originality  in  solution  and  flexibility  of 
expression. 


CHAPTEE  VIII 
THE  VALUE   OF  DRILL 

The  educational  pendulum  always  oscillates  between 
extremes.  Those  golden  means  which  the  practical  adminis- 
trator desires  are  seldom  discovered.  Much  of  the  arith- 
metic of  the  past  was  devoted  to  training  the  memory. 
Now  under  the  influence  of  the  movement  for  the  training 
of  the  higher  rational  processes  we  are  in  great  danger 
of  failing  to  reduce  to  an  automatic  basis  the  skills  formerly 
emphasized.  This  may  account  partly  for  the  criticism 
that  practical  men  urge  against  the  product  of  the  schools. 
Those  earlier  modes  of  instruction  that  called  for  an 
automatic  mastery  of  the  fundamentals  should  not  be  dis- 
carded without  a  hearing.  If  social  demands  are  to  be 
taken  as  a  criterion  for  judging  the  value  of  the  various 
subjects,  then  the  school  should  vigorously  insist  upon  a 
mastery  of  the  materials  and  processes  of  arithmetic.  This 
subject,  so  far  as  it  relates  to  common  community  life,  in- 
creases in  serviceableness  in  proportion  to  the  degree  to 
which  its  fundamental  processes  have  been  reduced  to  the 
plane  of  habit. 

Arithmetic:  A  Habit  Study 

Arithmetic  is  not  primarily  an  informational  subject; 
it  is  primarily  a  habit  subject.  By  this  statement  we  do 
not  mean  that  one  acquires  no  information  in  the  study  of 
arithmetic,  but  that  the  acquisition  of  facts  should  be  a' 
secondary  consideration.  Information  is  an  essential  out- 
come of  the  study  of  arithmetic,  but  the  habits  acquired 

92 


THE  VALUE  OF  DRILL  93 

are  elemental  or  fundamental.  We  would  not  eliminate 
from  arithmetic  problems  that  convey  information  about 
business  life ;  in  fact,  we  would  multiply  them.  But  in  do- 
ing so  we  would  not  neglect  to  give  a  proper  emphasis  to 
the  other  aspects  of  the  subject. 

Formal  Versus  Rational  Drill 

There  are  two  current  notions  as  to  the  manner  in  which 
habits  in  arithmetic  should  be  formed, — one  old  and  the 
other  new.  There  are  those  who  assert  that  children  should 
become  expert  in  handling  the  tables  before  they  put  the 
combinations  to  use  in  solving  problems.  About  twenty- 
five  years  ago  reading  was  taught  by  the  alphabetic  method. 
After  the  children  had  learned  to  repeat  the  alphabet  for- 
ward and  backward  and  could  combine  letters  into  mono- 
syllabic words,  the  teacher  permitted  them  to  read  a  few 
sentences.  Music  was  begun  by  singing  the  scale;  the 
children  were  told  that  if  they  learned  it  perfectly  some 
day  they  would  be  permitted  to  sing  some  songs.  In  draw- 
ing, the  children  first  learned  how  to  make  horizontal, 
vertical  and  oblique  lines ;  these  were  later  combined  into 
geometric  figures.  In  other  words  the  method  of  instruc- 
tion accepted  and  used  a  quarter  of  a  century  ago,  was  that 
the  habit  phases  of  each  subject  should  be  taught  and  mas- 
tered without  reference  to  their  use.  The  illusive  hope 
was  held  out  that  mastery  was  essential  because  the  date, 
the  place,  the  note,  or  the  number  combination,  might  be 
needed.  The  principal  motive  back  of  such  work  was  the 
demand  set  up  by  the  teacher;  the  manner  of  instruction 
was  intrinsically  uninteresting  as  few  devices  were  em- 
ployed. This  type  of  instruction  is  known  as  formal  drill. 
Its  chief  characteristic  is  that  facts  are  drilled  upon  in 
isolation. 


94  HOW  TO  TEACH  ARITHMETIC 

Rational  Drill 

In  contrast  to  this  we  have  to-day  what  is  known  as 
rational  drill.  It  is  an  attempt  to  swing  to  the  other  ex- 
treme. Teachers  are  urged  to  teach  by  projects  and  sit- 
uations that  are  close  at  hand.  Children  are  not  to  do 
anything  the  reason  for  which  they  do  not  understand  and 
which  they  do  not  consider  of  value  to  them.  Extreme 
as  this  theory  is  it  has  had  certain  beneficial  results.  To- 
day practically  every  good  teacher  of  reading  begins  with 
a  combination  of  words  that  make  sense  instead  of  with 
the  alphabet;  the  up-to-date  teacher  of  music  begins  with 
rote  songs ;  the  progressive  teacher  of  drawing  permits  the 
child  to  begin  with  a  free-hand  picture,  crude  and  imper- 
fect though  it  may  be;  and  the  modern  teacher  of  arith- 
metic begins  with  the  problems  that  are  related  to  the 
experiences  and  interests  of  children.  This  is  a  great  for- 
ward step  in  that  it  more  nearly  harmonizes  instruction  with 
the  rational  order  of  the  learning  process.  It  has  given 
interest  and  zest  to  the  daily  work  of  the  school/  School 
discipline  has  been  ameliorated  because  of  this  more  mod- 
'xern  way  of  teaching. 

We  are  urged  not  only  to  begin  each  subject  with  sit- 
uations and  problems  that  are  concrete  and  interesting, 
but  we  are  told  that  the  facility  children  need  in  reading 
will  be  acquired  by  multiplying  the  material  they  read; 
in  singing  by  having  them  sing  more  songs ;  in  drawing  by 
having  them  draw  more  pictures ;  in  arithmetic  by  having 
a  multitude  of  problems  solved.  In  other  words,  skill  in 
technique  is  to  be  acquired  by  increasing  the  number  of 
situations  in  which  it  normally  occurs.  For  example,  there 
comes  a  time  in  the  singing  exercises  when  attention  to  a 
bit  of  technique  becomes  necessary.  This  is  presented  and 
discussed  in  such  a  way  as  to  assist  the  children  in  under- 


THE  VALUE  OF  DRILL  95 

standing  it  and  in  interpreting  the  song.  This  bit  of 
technique  appears  the  next  day  in  that  same  song  and  on 
succeeding  days  in  other  songs.  Similarly  by  solving  many 
concrete  problems  the  student  in  arithmetic  becomes  more 
and  more  automatic  in  his  response  to  the  number  com- 
binations. 

The  critics  of  this  theory  maintain  that  it  requires  too 
much  time  to  produce  the  results  demanded  by  business, 
and  in  this  criticism  there  may  be  some  truth.  It  seems 
that  each  of  these  contending  theories  contains  an  element 
of  truth.  The  happiest  and  most  fruitful  combination 
would  probably  be  as  follows:  After  the  pupils  have 
learned  to  recognize  and  to  use  a  sufficiently  wide  range 
of  technical  phases  of  a  subject,  these  might  be  delib- 
erately taken  out  of  their  natural  setting  and  drilled  upon 
formally.  Such  formal  drill  would  have  a  distinct  ad- 
vantage over  the  old  type  of  formal  drill,  viz.,  the  chil- 
dren would  see  the  use  of  the  things  they  were  being 
drilled  upon. 

Illustration  of  Rational  Drill 

A  concrete  illustration  of  this  reconstructed  form  of  drill 
work  may  aid  in  clarifying  the  description.  The  sixth 
grade  children  in  the  training  school  of  a  western  normal , 
school  were  deficient«n  the  fundamental  operations.  It 
was  decided  to  subject  them  to  a  daily  drill  in  mental 
arithmetic.  The  teacher  who  was  able  to  employ  effi- 
ciently the  greatest  variety  of  devices  was  placed  in  charge 
of  the  work.  The  class  recited  in  a  room  adjoining  the  one 
in  which  they  studied.  The  recitation  in  mental  arithmetic 
began  the  instant  the  children  were  assembled  and  seated 
in  the  recitation  room.  The  teacher  did  not  say,  "Now, 
attention,  children;  we  are  going  to  have  a  little  drill  in 


96  HOW  TO  TEACH  AEITHMETIC 

mental  arithmetic  to-day.  Is  every  one  ready  ?  Now  listen 
carefully  while  I  state  the  first  example."  On  the  con- 
trary, the  children  understood  that  the  recitation  would 
start  the  instant  the  door  closed.  They  knew  the  teacher 
would  not  waste  several  precious  minutes  in  needless  pre- 
liminaries. To  be  ready  for  her  opening  statement  they 
were  leaning  forward  on  the  outer  edges  of  their  chairs, 
ready  to  start  just  as  a  runner  is  at  the  firing  of  a  pistol. 
The  recitation  moved  aggressively.  The  teacher  gave  such 
examples  as  "take  2,  multiply  it  by  2,  square  the  product, 
multiply  by  4,  take  one-half  that,  50  per  cent  of  that,  the 
square  root  of  that."  There  were  no  pauses  in  her  state- 
ment ;  she  uttered  these  statements  with  the  speed  one  uses 
in  ordinary  conversation.  The  very  moment  she  finished 
the  children  were  expected  to  be  ready  with  the  answer. 

After  a  few  weeks  those  who  were  interested  in  the  ex- 
periment checked  up  on  the  children  to  note  the  number 
of  mistakes  they  made  minute  by  minute.  It  was  noted 
that  few  mistakes  were  made  the  first  two  minutes,  they 
were  more  noticeable  during  the  third  minute,  very  notice- 
able the  fourth,  equalled  the  number  of  correct  answers 
the  fifth,  and  exceeded  the  number  of  correct  answers  the 
sixth  minute.  On  the  basis  of  this  evidence  it  was  decided 
that  only  three  minutes  a  day  would  be  devoted  to  mental 
arithmetic,  but  it  was  to  be  three  minutes  of  sixty  seconds 
each,  one  hundred  and  eighty  seconds  in  all.  The  experi- 
menters knew  full  well  that  this  amount  of  time  did  not 
agree  with  the  amount  set  aside  for  such  work  in  many 
courses  of  study. 

The  authors  are  convinced  that  the  time  that  can  be 
profitably  devoted  daily  to  oral  arithmetic  may  be  ex- 
tended beyond  three  minutes.  The  extension  of  time  de- 
pends upon  the  maturity  of  the  pupils  and  the  resource- 
fulness of  the  teacher.  A  great  variety  of  devices  is 


THE  VALUE  OF  DRILL  97 

necessary  if  children  are  to  be  kept  at  their  maximum  speed 
and  attention  longer  than  three  minutes. 

Results  of  This  Drill 

This  three-minute  recitation  became  one  of  the  "show 
things "  of  the  school.  The  children  looked  forward  to  it 
with  keen  pleasure.  In  order  that  they  might  excel  in  it 
they  learned  the  multiplication  tables  without  any  solicita- 
tion from  any  one  up  to  17,  '18,  19,  and  *20.  They  could 
tell  the  cube  of  16  as  quickly  as  the  average  teacher  can 
tell  the  cube  of  two.  *  The  writers  have  seen  groups  of  two, 
three,  and  four  of  them  in  the  corridor  of  the  building,  in 
vacant  rooms,  in  the  shade  of  trees  on  the  campus,  drilling 
each  other.  Perhaps  some  ' '  progressive ' '  may  criticize 
the  teacher  for  permitting  them  to  learn  the  multiplica- 
tion tables  beyond  twelve.  They  were  so  interested  in  the 
work  that  it  would  have  been  difficult  to  prevent  them. 
The  chief  reason  for  stopping  at  ten  or  twelve  is  that  it  is 
conventional. 

Whenever  work  of  this  type  is  advocated  there  is  always 
some  person  ready  to  sneer  at  it — a  person  who  remem- 
bers of  some  isolated  case  of  a  child  thus  drilled  who  never 
got  into  the  high  school,  or  who  failed  in  all  of  his  .other 
duties,  or  who  grew  weaker  in  the  reasoning  problems  in 
arithmetic.  We  want  to  assure  the  young  teacher  that  a 
three-minute  daily  drill  will  not  be  accompanied  by  any 
such  untoward  results. 

After  this  experiment  had  been  carried  on  for  some 
time  an  attempt  was  made  to  discover  the  effect  of  the 
drill  on  the  written  problems  of  the  text.  According  to 
the  testimony  of  the  critic  teacher  more  than  sixty  per 
cent  of  the  written  problems  of  the  text  had  become  mere 
oral  problems  to  the  children.  This  is  a  true  measure 
of  the  value  of  such  exercises.  Moreover,  the  facility  thus 


98  HOW  TO  TEACH  AEITHMETIC 

acquired  was  not  lost  during  vacation ;  it  carried  over  into 
each  of  the  two  succeeding  years. 

Although  this  experiment  lacks  some  of  the  scentific 
value  of  others  that  we  shall  describe  later,  still  it  is 
sufficiently  sound  in  all  particulars  to  show  the  proper 
correlation  of  rational  and  formal  drill,  and  also  to  show 
what  remarkable  results  may  be  achieved.  By  devoting 
three  minutes  a  day  to  rapid  mental  drill  a  teacher  or 
county  could  achieve  such  distinction  that  their  schools 
would  be  known  throughout  the  land.  It  hardly  needs  to 
be  said  that  personal  distinction  is  not  the  end  of  such 
work..  Three  minutes  a  day  will  bring  not  only  personal 
distinction;  it  will  put  the  pupils  in  secure  possession  of 
the  tools  needed  for  higher  mathematical  work;  it  will  so 
equip  them  as  to  break  down  one  of  the  criticisms  of 
the  outside  world. 

Some  Results  Attained  by  Brief  Drill 

In  this  connection  we  wish  to  relate  the  results  of  an- 
other successful  teacher  who  devoted  three  minutes  daily 
to  drill  work  in  arithmetic.  Sometime  near  the  close  of 
the  year,  three  examples  were  given  to  the  class  as  a  test. 
They  were : 

1.  Add    4587  2.  7654219x897=? 
8654  854796x2078=? 

4879  27864523x9376=? 

6875 

9897  3.  89765342 --97=? 
8546  4275897245 -f  789=?     . 

8465  987007648-654=? 

7699 
7967 
4567 


THE  VALUE  OF  DRILL 


99 


1.  (Coiit.)  7698 
8765 
7698 
8765 
7654 
6574 
5678 
9876 
5596 
8945 
7894 
4894 
4955 
7644 
8989 
4554 
6589 


The  three  examples  were  given  to 
pupils  whose  average  age  was  lOf  years. 
The  following  time  was  required  in 
solving  each  example: 


Time 
fastest 

Problem         pupil 

1 45  sec. 

2 110  sec. 

3  .  .  140  sec. 


First  25 
pupils 

finished  Average  time 
105  sec.  72  sec. 


270  sec. 
260  sec. 


184  sec. 
197  sec. 


We  are  not  attempting  to  justify  the  presence  of  such 
unusual  numbers  as  appear  in  these  examples.  We  recite 
the  facts  merely  to  show  what  may  be  accomplished  by  drill 
work  without  detrimental  results  to  the  other  subjects  of 
the  curriculum. 

Variations  of  Rational  Drill 

The  two  types  of  instruction  we  have  been  describing  in 
this  chapter  have  many  variations  when  applied  to  other 
phases  of  arithmetic.  One  of  the  most  common  contrasts 
is  revealed  by  the  following  incident: 

A  group  of  grade  teachers  was  granted  permission  to 
visit  a  system  of  schools.  Upon  their  return,  one  of  them 
described  a  visit  to  a  recitation  in  arithmetic.  She  said 
the  teacher  in  charge  had  the  day  before  assigned  four 


100  HOW  TO  TEACH  AEITHMETIC 

problems  for  the  class  to  solve.  When  the  recitation  was 
called,  the  teacher  spent  about  five  minutes  in  assisting 
those  children  who  had  not  been  able  to  solve  all  the  prob- 
lems to  discover  their  difficulties.  Then  she  gave  the 
remainder  of  the  time  to  the  solution  of  other  problems. 
The  critic,  commenting  upon  this  plan,  said,  "I  think  she 
should  have  had  a  child  solve  the  first  problem  upon  the 
board ;  another  child,  the  second  problem ;  a  third,  the 
third ;  and  a  fourth,  the  fourth.  After  they  were  all . 
placed  upon  the  board,  she  should  have  had  each  of  them 
explained. ? ' 

Here  we  have  two  types  clearly  differentiated.  How 
shall  we  determine  which  is  the  better  ?  Perhaps  we  should 
ask,  "What  is  the  test  of  efficiency  in  teaching  material  of 
this  kind  ?  The  test  of  efficiency  in  arithmetic  is  the  solu- 
tion of  problems.  If  the  children  are  able  to  solve  intelli- 
gently all  the  problems  in  a  proper  assignment,  it  is  prima 
facie  evidence  that  the  teaching  has  been  well  done ;  if  they 
solve  none  of  them,  it  is  first-hand  evidence  that  the  teach- 
ing has  been  poorly  done.  One  seldom  finds  either  of  these 
extremes.  What  he  does  find  is  that  some  of  the  children 
solve  all  of  the  problems,  some  none  of  them,  and  a  few 
some  of  them.  If  by  a  few  well-directed  questions  the 
teacher  is  able  to  make  those  who  have  totally  or  partially 
failed  conscious  of  the  character  of  their  difficulty,  she 
can  then  devote  the  remaining  time  to  other  matters.  In 
\  any  event,  she  discovers  those  who  are  in  need  of  individual 
attention. 

Much  time  is  wasted  in  having  children  go  ovver  and  over 
problems  that  they  have  already  demonstrated  their  ability 
/to  solve.     The  ability  of  children  to  solve  problems  is  a 
V  direct  measure  of  how  well  the  teaching  has  been  done.v 
If  the  time  could  be  saved  that  is  wasted  in  solving  prob- 
lems  previously   solved,   many   more   problems  might  be 


THE  VALUE  OF  DBILL  101 

solved  illustrating  the  application  of  the  same  principle; 
more  time  could  be  devoted  to  aggressive  and  effective 
drill  work;  and  more  attention  could  be  given  to  the 
teaching  of  the  methods  of  work  involved  in  those  new 
phases  of  the  subject  which  are  to  come  up  later. 

In  this  connection  the  writer  is  reminded  of  a  recitation 
in  a  third  grade  in  which  the  teacher  imagined  she  was 
doing  drill  work  in  arithmetic.  She  had  placed  upon  the 
blackboard  a  number  of  simple  examples  in  addition.  The 
class  was  called  to  "  Attention ";  one  pupil  was  called 
by  name;  he  ran  lightly  to  ttye  board;  the  teacher  read 
the  example  while  he  copied  it ;  he  performed  the  addition 
as  quickly  as  possible;  the  teacher  smilingly  nodded  her 
approval;  the  pupil  erased  the  work  and  ran  back  to  his 
seat.  This  performance  was  repeated  until  each  pupil  had 
solved  an  example.  The  recitation  began  on  time  and 
closed  on  time,  full  twenty  minutes  having  been  consumed. 

An  examination  of  this  shows  that  it  had  many  of  the 
marks  of  a  good  recitation.  Each  child  had  the  opportu- 
nity to  exercise  himself  physically — he  went  to  and  from 
the  board;  he  exercised  his  mental  activity,  for  he  solved 
an  example;  he  received  the  teacher's  nod  and  smile  of 
approval  for  the  work  done;  perfect  order  and  decorum 
prevailed  throughout. 

The  weakness  of  the  recitation  was  brought  out  in  the 
conversation  that  followed  at  the  rest  period.  The  teacher 
asked  the  supervisor,  who  witnessed  the  recitation,  what 
he  thought  of  it.  This  supervisor,  believing  that  his  chief 
business  was  to  improve  instruction,  asked  the  teacher  why 
she  did  not  send  one  pupil  to  the  place  where  she  had  all 
of  the  examples  copied,  and  then,  as  soon  as  the  class  had 
solved  the  first  example  and  some  pupil  had  stated  the 
answer,  have  the  boy  at  the  board  write  it  in  its  proper 
place.  The  supervisor  wanted  to  know  how  long  it  would 


1()1>  flOW  TO  TKACII    AIUTJIMKTIO 

h;ive  l;d<en  t.o  h;ivc  required  ;ill  MM-  pupils  1c>  solve  .'ill  of 
the  examples.  When  I  he  I  e;ichcr  s;iid,  "Oh,  not.  more  t.li;in 
•  •n  or  ei'dit  minute;;/'  the  :;iipervisnr  s;iid,  "Well?"  A 
moment.  Liter  lie  :;;iid,  "Well?"  more  inlerro«.';i  I  i  vcly  th;m 
before.  Then  MIC  le;icher  n;nv<-|y  ;;;ii«l7  "l>ut,  wh;il.  would  I 
fiavc  dour  vvilh  UK-  r«-:.|  of  Ihc  l.imr?" 

TlH*H(!    illusl  rjilioiiH   ;irc    not,   overdrawn    nor    l';ir  IVlclicd. 
Tliry   Jirn  d('H(jripl  i  vc  of  ;i    l\p<-  of   work    llud.  i",n\   r;i:;ily   he 
lound    in    iiuiny    phiccs   lod.'iy.      'rii«-:.«-    I  rndilioiuil    inclhodH 
an;    rvNpoiiNiblr    Tor    imi'-h    w.'i.sh-    in    lc;ic|iin^.      SiH-li    ;mli 
(jiinlcd     forms    oi'    iust  rud  ion     h;iv<-    l<  <l     lo    ron.spiciiously 
fvvron;r    nol  ion:;    in    r«'|»;;ird    lo    drill    work.      Sii''cc.,:;rid    <lrill 
worl-     IIH.III.;    :,h;irp,    (jiiick    (jllCHJ  ionilij<    Jind    immi'di;il.<-    r<- 
HpOMHCH;     it.    Nhould    IM-    UccoiupliHlicd    in    ;i    ::horl.    period    of 
I.  inir.       If    wr    climiimlr    tin-    non  csscnt  inls    IVoni    the    t< 
hooks  ;ind    tin-   lUM^'onoinicnl    nx-thods   From    MM-   instruction, 
we  sh;dl    li;ivr  ;in   ;ihund;i  nrr  of   time   to  dcvolc   lo   things  of 
;<    more  profit  n  I  ilr  ch;i  r;id<  r 

-i/  and 


Thr  two  pnm;iry  purposes  of  drill  work  ;m-  iiicrcn  cd 
Mccnrncy  ;md  iucrciiscd  :,prcd.  Acciirjicy  should  involve 
the  securing  oi'  tin-  correct  result,  in  the  short*-::!,  possible 
lime  with  the  minimum  e  x  pend  1  1  II  re  of  energy.  lli;ieeil 
r;iey  results  fi'om  ;d  tempt  in;»;  l.o  do  too  much,  from  work 
lh;d  i::  loo  dil'lieiill,  from  h;i:;ty  ;ilid  slipshod  methods, 
from  ;i  l';d::e  ;iltilude  oil  the  p;irl  of  le;iehers.  Te.'ieher.S 

soinet  i  me:;  eneonni^e    iiuieeuniey    hy    pr;i  i.,in<r    woi'k    lh;it    is 
wr-ong. 

Laws  of  I  I<i  hi  I,  Wonnulian 

Aeeur;iey  ;ind  speed  seem  to  he  closely  rehited  when  the 
Work  is  done  \\ith  the  m;i\imum  <le;«;ree  of  concent  \'\\\  ion. 
One  should  work  ;is  iiccui'.'ilely  .'is  he  e;m,  ;iml  ;is  r;ipidly 


TIIK    VAUIK   ()!•'    IHtlLL 

as    lie   can.      I » y    doiii;-;   I  his    lie    improves   simullaneously    in 
speed   and   ;ieeu  r;ie  y. 

Slowness  in   computation    is  <lue  lo  any  one  of  a  number 
of  factors.     The  most,  common  cause  is  1  he  lark  of  ndeqiintcA 
Iramiii"    in  eoiuil  in;1, a  ml   in   111:111  ipn  l;il  in;';  I  he   fundamental  / 
oper;il  ions.      When  a   <'hil<I   in  tin*  filth  or  si\lh   ;';rade   finds 
if  necessary   l.o  think   whal   .'»     7  ;ire,   he  is  either  backward 
or*    has    been    snhjeel    to    poor    instruction.      A    response    to 
such  combination  should  come  instantly.      The  lower  nerve 
centers  should    provide   the  corn-el    response. 

The  speed  of  I  he  children  is  sometimes  retarded  because 
they    are    required    to   work    too   lon^   upon   supposedly   con- 
crete   problems.      This    was    one    of    the    weaknesses    of    The/ 
Speer  method.      The  writer's  knew  a  school   in  which   pradi 
eally    no   problems   were  solved   <lurin;»;   the    first,    Tour  years 
of  the   pupil's  school    life   which    he  could    not,  demonstrate 
with    blocks  or§  pictures.      This  sometimes   resulted    in    keep-' 
in;r  children   so    Ion"    upon    the    plane   of  concrete    (Junk 
that     (hey    lost    or    failed    to    acquire    the    power    to    handle; 
abstract   operations.      Any    ^ood    device   of    this    kind    when 
used    to  extreme   becomes  an   evil. 

A  third  factor  which  retards  speed  in  computation  i.1- 
the  failure  to  make  the  operations  perfectly  simple  before 
drill  is  be^un.  At  the  outset  the  thin;':;  to  be  drilled  upon 
should  be  seen  and  understood  in  their  normal  Hit  nations. 
Then  attention  should  be  focused  upon  the  operation  itself. 
Mere  repetition  may  result  in  a  habit,  but  mere  repetition 
is  most  uneconomical. 

If  one  wishes  to  produce  number'  habits  with  the  le;i:;t 
expenditure  of  energy  and  lime,  he  should  permit  no 
i'tions,  the  correct  answer  must  be  ;»;iven  every  time. 
One  never  {'els  rijdil  result:;  by  praising  wron.^  answcrw!) 
Mvery  respoii.se,  every  reaction,  whether-  rijdit  or  wronj-;, 
tends  to  impress  itself  indelibly  upon  the  nervous  system. 


kcep-V 

ikintf'A 


104  HOW  TO  TEACH  ARITHMETIC 

Another  fact  to  be  remembered  is  that  mere  exhorta- 
tion on  the  part  of  the  teacher  does  not  produce  habits  in 
pupils.  It  is  literally  true  that  practice  makes  perfect, 
provided  the  practice  is  always  upon  the  correct  form.  In 
the  final  analysis,  one  acquires  particular  skill  only  by 
practicing  *the  things  which  give  that  skill. 

The  last  important  factor  to  be  mentioned  is  that  the 
periods  between  drills  should  be  gradually  lengthened,  and 
not  neglected.  If  we  drill  next  week  on  what  we  have 
acquired  this,  then  next  month,  then  four  months  from 
now,  then  next  term,  then  next  year,  and  so  on,  there  is 
little  danger  of  children  leaving  school  without  acquiring 
many  desirable  number  habits.  It  is  for  this  very  reason 
that  texts  should  provide  more  liberally  for  reviews.  Re- 
views that  call  for  the  mastery  and  fixing,  as  well  as  the 
wider  application  of  facts  and  principles,  are  essential  to 
scholastic  attainment  in  every  field,  not  alone  in  arithmetic. 

Scientific  Studies  of  Drill 

One  of  the  earliest  studies  of  habit  formation  in  arith- 
metic is  that  by  Professor  E.  L.  Thorndike,  on  "Practice 
in  the  Case  of  Addition77  (American  Journal  of  Psychol- 
ogy, 21;  483-486).  The  test  was  made  with  nineteen 
university  students,  eight  men  and  eleven  women.  These 
added  daily  for  seven  days  forty-eight  columns  of  ten  num- 
bers each.  No  column  contained  any  O's  or  1's.  The  time 
of  each  addition  was  kept  in  seconds,  and  a  record  was 
kept  for  the  number  of  correct  and  incorrect  results.  The 
experiment  showed  four  things:  (1)  that  improvement  in 
speed  and  accuracy  was  about  equal;  (2)  the  fact  that 
adults  can  improve  in  a  skill  of  this  kind  is  good  evidence 
that  improvement  in  any  intellectual  trait  is  mainly  the 
result  of  special  training;  (3)  that  practice  improve- 


THE  VALUE  OF  DRILL  105 

ment  is  greatest  when  one  works  up  to  his  limit  in  com- 
petition with  his  own  past  record  (this  is  the  right 
incentive  for  special  drills  in  regular  school  work) ;  (4) 
that  variability  between  individuals  decreases  with  drill. 
An  extensive  study  of  the  effect  of  drill  in  the  funda- 
mental operations  of  arithmetic  -was  made  by  Mr.  Brown, 
one  of  the  co-authors  of  this  book. 

How  the  Investigation  was  Conducted 

Tests  were  given  in  the  sixth  grades  of  three  different 
public  school  systems  and  in  the  sixth  grade  of  a  large 
private  school.  The  total  number  of  cases  recorded  in  this 
study  was  222 ;  of  these,  110  were  boys  and  112  were  girls. 

The  three  public  schools  examined  are  in  the  Central 
West.  City  C  has  a  population  of  seven  thousand ;  City  M, 
.of  twelve  thousand;  and  City  D,  of  thirty  thousand.  The 
private  school  is  in  New  York  City. 

The  effects  of  the  drill  in  fundamentals  were  shown  by  a 
comparison  of  sections  subjected  to  the  drill  with  sections 
of  equal  size  and  approximately  equal  ability  not  subjected 
to  the  drill  but  otherwise  undergoing  the  same  arithmetical 
instruction.  The  object  was  to  determine  the  improvement 
made  by  the  drill  class  upon  its  previous  record  and  the 
improvement  made  by  the  non-drill  class  upon  its  pre- 
vious record. 

In  a  given  class  the  tests  were  conducted  at  the  same 
hour  of  the  school  day,  in  order  to  eliminate  the  time  factor 
as  far  as  possible. 

Immediately  after  the  first  test  was  given  in  each  school, 
half  of  the  classes  examined  in  each  city  were  given  five 
minutes'  drill  each  day  upon  the  fundamental  operations 
in  arithmetic, — addition,  subtraction,  multiplication,  and 
division.  The  first  five  minutes  of  the  recitation  period  in 


106  HOW  TO  TEACH  AEITHMETIC 

arithmetic  were  devoted  to  the  drill  work.  The  drill  was 
partly  oral  and  partly  written,  and  the  time  was  about 
evenly  distributed  among  the  four  operations. 

No  special  instructions  were  given  to  the  teachers  in 
charge  of  the  drill  sections,  except  that  they  were  to  empha- 
size both  speed  and  accuracy  in  the  four  operations,  and 
were  to  cover  the  same  daily  assignments  in  the  text-books 
as  the  class  that  had  no  drill.  The  teachers  of  the  non-drill 
classes  were  asked  to  give  no  formal  drill  upon  any  of  the 
four  fundamental  operations  during  the  time  that  this 
investigation  was  in  progress.  These  instructions  were 
'  carefully  observed  by  the  teachers. 

The  drill  classes  in  each  city  were  able  to  cover  the  same 
subject-matter  of  the  text  as  the  non-drill  classes  of  that 
city.  No  special  tests  were  given  to  determine  the  com- 
parative excellence  of  the  text-book  work,  but  in  every  case 
the  teacher  in  charge  of  a  drill  class  reported  that  five 
minutes  devoted  to  drill  at  the  beginning  of  each  recitation 
seemed  to  act  as  a  mental  tonic.  It  seemed  to  energize  the 
pupils  and  to  make  them  keen  for  the  text -book  work  that 
was  to  follow.  All  teachers  of  drill  classes  reported  an 
improvement  in  text-book  work. 

Formal  drill  work  on  the  four  fundamental  operations 
had  not  been  given  prior  to  this  investigation  in  any  of 
the  sixth  grades  examined.  Whatever  marked  changes 
occurred  in  all  of  the  drill  sections  that  did  not  occur  in 
the  non-drill  sections  may  reasonably  be  attributed  to  the 
results  of  the  special  drill. 

Results  of  the  Drill 

If  the  number  of  problems  worked  in  each  test  may  be 
taken  as  a  measure  of  the  speed  of  the  pupils,  the  drill 
class  increased  its  speed  by  16.9  per  cent  and  the  non-drill 
class  by  6.4  per  cent. 


THE  VALUE  OF  DRILL  107 

Since  practically  all  of  the  pupils  finished  at  least  the 
first  six  problems  in  each  test,  a  comparison  of  the  records 
made  on  these  six  problems  will  give  a  basis  for  determin- 
ing the  relative  accuracy.  Measured  by  this  standard,  the 
drill  class  made  a  gain  of  11.7  per  cent  in  accuracy, 
whereas  the  non-drill  class  actually  lost  in  accuracy  (-1.8 
per  cent). 

The  largest  gain  made  by  the  drill  class  was  in  division, 
34.2  .per  cent,  which  was  more  than  twice  the  gain  made  in 
division  by  the  non-drill  class,  15.4  per  cent. 

If  we  compare  the  gain  made  by  the  drill  class  upon  its 
own  record  with  the  gain  made  by  the  non-drill  class  upon 
its  own  record,  we  find  that  the  following  results  were 
attained : 

Drill  class  gained  2.64  times  as  much  as  non-drill  class 
on  problems  worked.  ' 

Drill  class  gained  2.72  times  as  much  as  non-drill  class 
in  addition. 

Drill  class  gained  2.68  times  as  much  as  non-drill  class 
in  subtraction. 

Drill  class  gained  2.21  times  as  much  as  non-drill  class 
in  multiplication. 

Drill  class  gained  3.13  times  as  much  as  non-drill  class 
in  division. 

Drill  class  gained"  2.57  times  as  much  as  non-drill  class 
in  total  number  of  points. 

The  drill  classes  made  from  two  and  one-fifth  to  three 
and  one-tenth  times  as  much  improvement  as  the  non-drill 
classes.  It  is  worthy  of  note  that  the  average  age  in  the 
drill  classes  was  exactly  the  same  as  the  average  age  in  the 
non-drill  classes,  being  twelve  and  two-tenths  years  in  each 
case. 

In  the  following  table  the  first  test  was  given  before  the 
drill  was  begun,  the  second  test  was  given  immediately 


108  HOW  TO  TEACH  ARITHMETIC 

after  the  thirty  days'  drill,  and  the  third  test  was  given 
on  the  first  day  of  the  fall  term,  after  a  vacation  of  twelve 
weeks : 

A  COMPARISON  OF  THE  EESULTS  OF  THE  THIRD  TEST  WITH  THE  FIRST 
AND  SECOND. 

('*!"  indicates  combined  drill  sections.   "II"  the  non-drill  sections.) 
I  did  26.4  per  cent  better  than  on  first  test  and  4.1  per  cent  better 

than  on  second  test  in  number  of  problems  worked. 
II  did  9.8  per  cent  better  than  on  first  test  and  same  as  on  second 

test  in  number  of  problems  worked. 
I  did  25.4  per  cent  better  than  on  first  test  and  6  per  cent  poorer 

than  on  second  test  in  addition. 
II  did  7.7  per  cent  better  than  on  first  test  and  3.7  per  cent  poorer 

than  on  second  test  in  addition. 
I  did  46.2  per  cent  better  than  on  first  test  and  6.7  per  cent  better 

than  on  second  test  in  subtraction. 
II  did  20.4  per  cent  better  than  on  first  test  and  6.4  per  cent  better 

than  on  second  test  in  subtraction. 
I  did  31.3  per  cent  better  than  on  first  test  and  1.5  per  cent  better 

than  on  second  test  in  multiplication. 
II  did  11.1  per  cent  better  than  on  first  test  and  2.2  per  cent  poorer 

than  on  second  test  in  multiplication. 
I  did  36.7  per  cent  better  than  on  first  test  and  7.3  per  cent  better 

than  on  second  test  in  division. 
II  did  11.1  per  cent  better  than  on  first  test  and  2.8  per  cent  poorer 

than  on  second  test  in  division. 
I  did  31.7  per  cent  better  than  on  first  test  and  0.2  per  cent  poorer 

than  on  sproml  test  in  total  points.  * 
II  did  12.16  per   cent  better   than   on   first   test   and   2.29   per   cent 

poorer  than  on  second  test  in  total  points. 
I  did  5.2  per  cent  better  than  on  first  test  and  0.6  per  cent  poorer 

than  on  second  test  in  first  six  problems. 

II  did  3.7  per  cent  poorer  than  on  first  test  and  1.3  per  cent  poorer 
than  on  second  test  in  first  six  problems. 

The  results  of  the  third  test  indicated  that  the  supe- 
riority of  the  drill  class  was  maintained  over  the  vacation 
period.  The  "period  of  hibernation"  served  to  increase 


THE  VALUE  OF  DKILL  109 

the  speed,  while  those  who  had  not  had  the  advantage  of 
the  drill  worked  no  faster  than  on  the  second  test.  The 
non-drill  section  either  made  no  improvement  or  did  worse 
than  on  the  second  test  in  everything  except  subtraction. 

The  conclusions  reached  by  Mr.  Brown1  were  corrobo- 
rated in  all  essential  particulars  by  Dr.  T.  J.  Kirby  in  a 
study  entitled  "Practice  in  the  Case  of  School  Children," 
published  by  Teachers'  College  Bureau  of  Publications, 
1913. 

Mr.  Kirby  also  found  that  a  brief  drill  period  produces 
better  results  than  a  longer  period. 

No  investigation  has  yet  been  made  to  determine  the 
relative  efficiency  of  drill  periods  from  one  to  ten  or  fifteen 
minutes,  or  whether  the  same  length  of  period  is  best  for 
each  of  the  fundamental  operations. 

i  A  detailed  account  of  Mr.  Brown  Js  investigation  is  given  in  the 
Journal  of  Educational  Psychology,  November  and  December,  1912. 


CHAPTER  IX 

WASTE  IN  ARITHMETIC 

This  is  an  age  of  great  commercial  and  industrial 
activity.  The  invention  and  improvement  of  numerous 
labor-saving  devices,  the  improved  facilities  for  transpor- 
tation and  communication,  and  the  significant  discoveries 
in  numerous  fields  of  scientific  research  have  revolutionized 
commercial,  industrial,  and  economic  conditions  in  the 
United  States  within  the  last  few  decades.  Efficiency  has 
become  the  watchword,  and  efforts  are  being  made  to  elimi- 
nate all  possible  waste  of  time,  energy,  and  material.  The 
efficiency  engineer  is  a  product  of  our  modern  conditions. 

The  progressive  educator  of  to-day  is  interested  in  any 
investigation  seeking  to  establish  standards  of  efficiency 
that  may  be  applied  to  the  schools.  Within  recent  years 
several  school  systems  have  been  carefully  examined  and 
their  efficiency  has  been  judged  by  experts  who  had  no 
immediate  or  personal  interest  in  them.  In  so  far  as  the 
equipment,  organization,  and  instruction  of  the  schools  can 
be  standardized  and  the  efficiency  measured,  such  investi- 
gations, when  conducted  with  a  genuine  desire  to  learn 
existing  conditions,  are  of  real  value. 

It  is  a  comparatively  easy  matter  to  establish  standards 
of  excellence  for  many  material  objects,  and  to  classify  more 
or  less  accurately  the  quality  of  a  given  product  when 
compared  with  a  standard  product  of  the  same  kind.  It  is 
vastly  more  difficult,  if  not  impossible,  to  set  up  exact 
standards  of  efficiency  for  all  mental  products.  Some 
phases  of  school  organization,  equipment,  and  management 

110 


WASTE  IN  ARITHMETIC  HI 

can  be  measured  with  considerable  accuracy  when  com- 
pared with  recognized  standards.  Some  of  the  products 
of  the  school  cannot  as  yet  be  measured  as  accurately  as  we 
wish,  but  this  should  not  deter  us  from  applying  standards 
of  greater  or  less  refinement  to  those  that  can  be  measured. 
The  progressive  educator  will  look  with  favor  upon  all 
attempts  that  are  made  to  refine  the  standards  of  measure- 
ment now  in  use,  and  will  welcome  the  establishment  of 
other  appropriate  standards  for  the  measurement  of  school 
activities  and  products.  The  teacher  should  constantly 
seek  to  establish  a  better  balance  between  effort  and  result, 
a  closer  adjustment  of  means  to  ends.  As  far  as  is  possible, 
standards  for  the  evaluation  of  school  activities  should  be 
used.  Any  investigation  of  subject-matter  or  of  method 
in  a  given  field  that  results  in  a  discovery  of  ways  and 
means  of  securing  a  better  balance  between  effort  and 
result  is  justified. 

This  chapter  is  devoted  to  a  consideration  of  the  extent 
to  which  waste  can  be  eliminated  in  the  teaching  of  arith- 
metic. The  topic  will  be  considered  from  the  point  of 
view  of  both  the  subject-matter  and  the  methods  of 
instruction. 

Before  a  manufacturer  can  determine  the  percentage  of 
waste  involved  in  making  a  given  product,  he  must  have 
in  mind  the  exact  product  that  is  to  be  made.  If  he  wishes 
to  determine  the  percentage  of  waste  material,  he  must 
investigate  the  extent  to  which  the  raw  material  balances 
the  output.  There  must  be  a  consideration  of  means  and 
of  ends.  A  better  adjustment  of  means  to  ends  always 
means  an  economic  gain.  If  we  wish  to  investigate  the 
sources  of  waste  in  arithmetic  we  must  first  establish  more 
or  less  clearly  the  aims  in  teaching  the  subject.  Unless  the 
goals  which  are  striven  for  are  known,  the  degree  to  which 
current  practice  enables  us  to  reach  the  goals  cannot  be 


112  HOW  TO  TEACH  ARITHMETIC 

determined.     The  degree  of  closeness  of  approach  to  the 
ideals  to  be  reached  is  a  factor  of  great  importance. 

We  would  justly  condemn  the  manufacturer  who  had 
no  clear  idea  of  what  he  was  attempting  to  produce.  In 
the  industrial  world  an  efficient  manager  knows  what  he 
is  attempting  to  do,  and  he  sees  the  necessity  for  each  step 
in  the  process.  Each  workman  strives  to  accomplish  a 
definite  end.  The  part  of  the  work  that  he  does  may  be 
but  a  small  part  of  that  necessary  to  make  the  finished 
product,  but  it  is  the  duty  of  each  one  to  do  his  part  of 
the  work  to  the  best  of  his  ability,  whether  it  be  the  turning 
of  a  spool  or  the  adjusting  of  a  watch.  It  is  not  easy  to 
determine  what  the  final  output  should  be  in  education,  or 
what  is  the  best  method  of  securing  the  desired  results. 

Aims  of  Education 

The  aims  of  education  are  numerous,  and  the  methods  of 
attaining  the  aims  must  vary  with  the  child.  Some  promi- 
nent writers  on  education  enumerate  five  or  six  aims.1  A 
variety  of  individuals  makes  necessary  a  variety  of  aims. 
Aimless  teaching  is  wasteful  teaching.  Every  teacher  should 
have  some  working  statement  of  aim,  but  this  may  vary. 

Educators  are  not  agreed  in  regard  to  all  of  the  ideals 
to  be  striven  for  in  the  teaching  of  arithmetic,  and  it  is  not 
probable  that  complete  agreement  will  be  reached  upon  this 
point.  Teachers  of  history,  geography,  grammar,  and  the 
industrial  arts  are  not  in  agreement  in  regard  to  the  ideals 
to  be  attained  in  their  respective  subjects.  Numerous  rea- 
sons for  the  teaching  of  arithmetic  have  been  advanced,  but 
most  people  will  agree  that  in  the  main  the  subject  has  a 
two-fold  justification.  It  is  taught  because  of  its  practical 

iSee  Bagley,  ''Educative  Process,"  pp.  40-65;  O'Shea,  "Educa- 
tion as  Adjustment, "  ch.  4  and  5. 


WASTE  IN  AKITHMETIC  113 

bread-and-butter  value,  and  because  of  the  opportunity 
that  it  affords  for  training  the  pupil  in  concise,  logical 
thinking.  Arithmetic  is  by  no  means  the  only  subject  that 
serves  these  two  purposes,  but  that  it  does  serve  them  in  a 
distinctive  way,  few  will  deny.  No  one  questions  the  neces- 
sity of  a  mastery  of  certain  fundamental  number  relations ; 
all  admit  that  a  child  must  acquire  a  certain  mastery  of 
quantitative  relationships.  It  is  not  an  easy  matter  to 
evaluate  the  mental  training  that  a  pupil  gets  from  arith- 
metic that  he  does  not  get  equally  from  other  subjects.  The 
pupil  is  probably  more  conscious  in  arithmetic  than  in  any 
other  subject  in  the  grades  that  he  has  come  into  contact 
with  certain  truth.  When  the  mathematician  Laisant  was 
asked  the  relative  importance  of  the  utility  and  the  culture 
value  of  arithmetic,  he  replied  that  it  is  like  asking  which  is 
the  more  important,  sleeping  or  eating, — the  loss  of  either  is 
fatal.  The  teacher  who  recognizes  but  one  of  these  aims  is 
not  teaching  most  effectively.  "The  practical  side  must 
concede  to  the  disciplinary  side  by  having  its  processes 
understood  when  they  are  presented,  even  though  the  child 
is  not  called  upon  to  remember  the  reasoning.  The  disci- 
plinary side  must  concede  to  the  practical  by  selecting  its 
topics  in  such  a  way  as  to  give  no  false  notions  of  business, 
and  as  to  encourage  the  pupils  to  take  an  active  interest  in 
the  quantitative  side  of  the  world  about  them."1 

Specific  Aim  of  Arithmetic  in  Each  Grade 

The  teacher  should  recognize  not  only  the  general  pur- 
poses for  which  arithmetic  is  taught,  but  he  should  have 
clearly  in  mind  the  specific  aim  of  the  work  in  each  of  the 
grades  in  which  he  teaches.  It  is  not  wise  to  attempt  to 
draw  a  fine  distinction  between  the  aims  of  the  work  in 

i Smith,  "The  Teaching  of  Arithmetic,7'  p.  21. 


HOW  TO  TEACH  ARITHMETIC 

each  of  the  grades.  These  aims  will  be  influenced  more  or 
less  by  local  and  individual  factors,  but  in  general  certain 
purposes  may  be  considered  as  of  prime  importance  in  the 
respective  grades. 

The  object  of  the  work  of  the  first  and  second  grades  is 
to  aid  the  pupils  to  image  clearly  the  objects  and  groups 
of  objects  in  proper  number  relations;  to  make  clear  and 
definite  quantitative  imagery.  This  may  be  accomplished 
through  emphasis  upon  counting,  upon  addition  and  sub- 
traction, upon  simple  estimates  and  comparisons;  games, 
rhymes,  drawing,  construction  work,  and  the  like. 

It  is  the  specific  object^of  the  third  and  fourth  school 
years  to  make  the  pupil  proficient  in  addition,  subtraction, 
multiplication,  and  division  with  whole  numbers  and  cer- 
tain fractions,  as  a  basis  for  the  work  that  is  to  follow. 
Emphasis  should  be  placed  upon  accuracy  and  speed  in 
both  oral  and  written  work.  By  the  use  of  simple  prob- 
lems adapted  to  the  experience  of  the  pupil,  there  should 
be  developed  the  ability  to  interpret  simple  number  rela- 
tions and  to  reason  from  simple  data.  The  pupil's  appre- 
ciation of  quantitative  relationships  in  the  life  about  him 
should  be  considerably  developed  in  these  grades,  through 
continued  use  of  measurement  and  estimates,  and  a  study 
of  the  simple  tables  of  compound  numbers. 

In  the  fifth  and  sixth  grades  the  mechanics  of  the  funda- 
mental processes  with  integers  and  fractions  should  be 
thoroughly  mastered.  There  should  also  be  increased  em- 
phasis upon  the  solution  of  problems.  Considerable  time 
should  be  devoted  to  the  selection  of  appropriate  processes 
for  solving  problems.  Every  effort  should  be  made  to 
develop  independent  thought  by  strengthening  the  judg- 
ment in  the  selection  of  correct  processes.  The  use  of 
appropriate  checks  should  become  habitual  in  these  grades, 
and  this  will  do  much  to  increase  confidence  and  to  bring 


WASTE  IN  ARITHMETIC  115 

pride  in  accomplishment  that  is  so  desirable.  Problems  in 
these  and  all  other  grades  should  be,  as  far  as  possible, 
within  the  pupil's  experience.  The  data  of  geography, 
history,  science,  and  manual  training  may  be  utilized  to  a 
considerable  extent. 

The  pupil  who  passes  the  age  of  eleven  or  twelve  without 
the  ability  to  perform  the  fundamental  operations  with 
facility  and  a  relatively  high  degree  of  accuracy  is  quite 
likely  to  be  handicapped  in  these  respects  throughout  life. 
Teachers  of  the  fifth  and  sixth  grades  have  a  great  respon- 
sibility upon  them,  and  the  pupil's  success  in  arithmetic  in 
the  following  grades  depends  to  no  small  degree  upon  the 
results  that  are  secured  in  these  grades.  Many  a  child  is 
sent  from  these  grades  a  cripple  in  his  work  in  arithmetic. 
Whenever  possible,  individual  attention  to  specific  needs 
should  be  given,  and  much  loss  of  time  in  later  grades  may 
thus  be  avoided. 

It  is  the  purpose  of  the  arithmetic  of  the  seventh  and 
eighth  grades  to  give  a  mastery  of  percentage  and  its 
modern  applications,  of  mensuration,  ratio  and  proportion, 
and  of  square  root ;  to  afford  a  thorough  and  comprehen- 
sive review  of  the  entire  subject  of  arithmetic ;  and  to  give 
the  pupil  some  knowledge  of  the  elements  of  algebra  and 
geometrical  construction.  In  these  grades  especially,  the 
larger  aspects  of  social,  industrial,  and  economic  life  should 
be  emphasized.  The  solution  of  numerous  problems  and 
the  application  of  appropriate  checks  should  be  continued, 
and  every  effort  should  be  made  to  develop  independence 
of  judgment  and  confidence  in  results. 

Elimination  of  Topics 

The  arithmetic  of  the  recent  past  included  several  topics 
which  should  have  been  eliminated  because  they  no  longer 
served  any  practical  end.  Traditions  of  the  school  have 


116  HOW  TO  TEACH  AEITHMETIC 

always  exercised  a  powerful  influence  in  retaining  topics 
in  arithmetic  after  their  period  of  usefulness  has  passed. 
Certain  topics,  like  certain  organs  in  the  human  body,  tend 
to  persist  after  their  original  functions  have  been  outgrown. 

In  former  years,  the  defense  offered  for  the  retention  of 
any  topic  in  arithmetic  that  was  no  longer  of  practical 
value  was  that  it  possessed  a  disciplinary  value,  and  this 
was  thought  to  justify  it.  To-day  we  recognize  both  the 
practical  and  the  disciplinary  value  of  arithmetic,  but  we 
do  not  believe  that  any  topic  should  be  retained  merely 
because  of  its  disciplinary  value.  We  seek  the  maximum 
of  mental  discipline  in  the  topics  that  have  some  practical 
value.  If  a  practical  topic  is  properly  presented,  the  disci- 
plinary value  will  be  obtained  from  it.  Both  insight  and 
skill  may  be  acquired  from  the  study  of  such  a  topic. 

One  of  the  great  sources  of  waste  in  arithmetic  to-day  is 
due  to  the  fact  that  we  have  not  yet  eliminated  all  topics 
that  find  no  application  in  present-day  practice.  The 
established  traditions  of  the  schools,  the  tendency  of  teach- 
ers to  teach  as  they  were  taught,  and  the  fascination  of 
some  of  the  old  types  of  problems,  tend  to  perpetuate  the 
obsolete.  -It  is  wasteful  to  teach  any  topic  in  the  grades 
that  the  pupil  will  not  use  in  some  way,  either  outside  of 
the  classroom  or  in  the  school  work  that  comes  later.  Even 
in  the  latter  case  a  topic  should  not  usually  be  presented 
until  some  use  is  to  be  made  of  it.  TO  say  that  the  mind  of 
the  pupil  is  developed  by  the  study  of  an  obsolete  topic  does 
not  justify  its  retention,  because  the  same  or  a  greater 
development  may  be  secured  by  the  study  of  topics  of 
genuine  practical  value.  Any  course  in  arithmetic  that 
does  not  look  forward  to  the  life  of  the  twentieth  century 
rather  than  back  to  the  nineteenth  is  wasteful.  Any  course 
that  does  not  seek  to  apply  arithmetic  to  the  problems  of 
daily  life  is  not  fulfilling  its  full  purpose. 


WASTE  IN  ARITHMETIC  117 

What  Topics  Should  Be  Eliminated 

The  demands  of  the  present  insist  that  the  topics  enu- 
merated below  be  omitted  from  a  course  in  arithmetic  in 
the  grades : 

1.  Greatest  common  divisor  and  least  common  multiple 
of  all  numbers  not  readily  factored,  and  all  work  involving 
the  Euclidean  method.     The  terms  of  fractions  to-day  are 
relatively  small  and  easily  factored,  or  the  fractions  are 
expressed  decimally.    Prior  to  the  beginning  of  the  seven- 
teenth century,  business  practice  demanded  a  knowledge 
of  the  long-division  form -of  the  greatest  common  divisor. 
It  is  not  improbable  that  the  subject  of  greatest  common 
divisor  and  least  common  multiple  will  be  omitted  entirely 
from  courses  in  arithmetic  in  a  few  years.     The  common 
fractions  taught  should  be  those  actually  used  in  business 
and  in  practical  life. 

2.  .All  obsolete  tables  in  denominate  numbers  and  all 
tables  that  are  of  use  to  the  specialist  only.    Troy  and 
Apothecaries  weight  are  not  of  importance  to  most  people. 

3.  All  problems  dealing  with  compound  numbers  of  more 
than  two  or  three  denominations  and  all  reduction  from 
Troy  to  Apothecaries  weight  should  be  omitted. 

4.  All  work  in  circulating  ^decimals  should  be  omitted; 
the  topic  should  be  studied  as  a  part  of  infinite  series 
in  algebra. 

5.  All  applications  of  percentage  that  do  not  conform  to 
present-day  practices  should  be  omitted.     True  discount 
should  not  be  taught.    The  numerous  state  rules  on  partial 
payment  should  receive  no  consideration  unless  they  are 
in  general  use  in  the  community.    The  types  of  negotiable 
paper  should  be  those  in  common  use  among  business  men. 

Equation  of  payments  is  not  of  practical  value  to-day 
to  anyone  except  a  few  specialists  in  foreign  commercial 


118  HOW  TO  TEACH  ARITHMETIC 

transactions.      Annual  interest   should  not  receive  much 
emphasis. 

6.  Cube  root  should  not  be  taught  in  the  grades. 

7.  Progressions  have  no  practical  value  in  arithmetic. 
The  theory  of  progressions  is  distinctly  algebraic. 

8.  •  Compound  proportion  has  been  largely  replaced  by 
unitary  analysis.     Even  simple  proportion  is  of  less  im- 
portance  than   formerly,   because   of   increased   emphasis 
upon  the  simple  equation  and  analysis. 

9.  Problems  which  require  long  and  involved  solutions 
or  answers  should  be  omitted.     Those  listed  below,  copied 
from  an  old  arithmetic,  are  extreme  illustrations  of  this. 

Typical  Problems  from  Musgrove  and  Wright's  "British 
American  Commercial  Arithmetic,"  published  in  1866, 
Toronto,  Canada. 

1.  Simplify 


2.  Reduce  the  common  fraction  .-fa  to  a  decimal. 

Ans.    .020408163265306122448979591836734693877551 

3.  A,  in  a  scuffle,  seized  on  f  of  a  parcel  of  sugar  plums, 
B  caught  f  of  it  out  of  his  hands,  and  C  laid  hold  on  y\ 
more;   D  ran  off  with  all  A  had  left,  except  £,  which  E 
afterwards  secured  slyly  for  himself;  then  A  and  C  jointly 
set  upon  B,  who,  in  the  conflict,  let  fall  \  he  had,  which 
were  equally  picked  up  by  D  and  E.    B  then  kicked  down 
C's  hat,  and  to  work  they  all  went  anew  for  what  it  con- 
tained; of  which  A  got  J,  B  ^,  D  f,  and  C  and  E  equal 
shares  of  what  was  left  of  that  stock.    D  then  struck  f  of 
what  A  and  B  last  acquired,  out  of  their  hands  ;  they,  with 
some  difficulty,  recovered  f  in  equal  shares  again,  but  the 
other  three  carried  off  -J-  apiece  of  the  same.     Upon  this, 


WASTE  IN  ARITHMETIC  119 

they  called  a  truce,  and  agreed  that  the  ^  of  the  whole  left 
by  A  at  first  should  be  divided  equally  among  them;  how 
many  plums  after  this  distribution  had  each  of  the  com- 
petitors ? 

Ans.    A  had  2863 ;  B,  6335 ;  C,  10294 ;  and  E,  4950. 

Enriching  the  Course 

Much  waste  has  been  eliminated  from  the  course  in  arith- 
metic by  the  omission  of  obsolete  topics,  but  great  care 
must  be  exercised  to  insure  that  some  equally  useless  topics 
are  not  substituted  for  those  that  have  been  eliminated 
We  are  attempting  to-day  to  enrich  our  modified  curric- 
ulum by  introducing  numerous  problems  from  practical 
life.  "We  are  seeking  to  secure  a  better  mastery  of  the 
fundamental  operations  through  greater  emphasis  on 
systematic  drill  and  through  greater  emphasis  on  oral 
arithmetic. 

Unnecessary  Computation 

There  is  waste  of  time  and  of  effort  in  arithmetic  when- 
ever a  process  is  carried  further  than  the  data  upon  which 
it  is  based  would  justify.  No  result  in  mathematics  can 
be  more  accurate  than  the  data  upon  which  it  is  based. 
Many  pupils  never  appreciate  the  absurdity  of  finding  the 
interest  on  a  given  sum  to  within  a  millionth  part  of  a 
cent,  or  of  finding  the  circumference  of  a  carriage  wheel  to 
within  a  millionth  of  an  inch.  For  most  practical  compu- 
tations a  result  computed  to  two  or  three  decimal  places  is 
sufficiently  accurate. 

Time  is  wasted  because  of  the  performance  of  unneces- 
sary operations.  It  is  wise  never  to  multiply  until  you  are 
forced  to,  and  never  divide  until  you  are  obliged  to.  The 
following  problem  will  illustrate  this  point:  Required 


120  HOW  TO  TEACH  ARITHMETIC 

to  find  the  radius  of  a  circle  equal  to  the  combined  area  of 
two  circles  of  radii  6  and  8  inches,  respectively. 

Necessary  Work: 

The  area  of  the  required  circle  =  3  6-n-  +  64?r  = 


The  radius  of  any  circle  -\\— 

\   7T 

The  radius  of  the  required  circle  =  ,  /  1007r  = 


In  solving  such  a  problem  many  pupils  would  multiply 
both  the  36  and  64  by  the  numerical  value  of  ?r,  then  add 
the  products.  This  multiplication  is  unnecessary. 

Cancellation  frequently  enables  one  to  save  time  in  the 
solution  of  a  problem. '  For  example : 

1.  If  29  bushels  of  potatoes  sell  for  $20.88,  what  will  31 
bushels  sell  for  at  the  same  rate  ? 


SOLUTION  : 

31  x  $20.88 


-$22.32 


29 

Time  Devoted  to  Arithmetic 

For  many  years  arithmetic  was  the  foremost  study  in 
the  curriculum  in  determining  the  pupil's  standing.  It  is 
still  an  important  subject  in  this  respect,  but  its  position 
of  preeminence  has  been  taken  by  English  or  Reading. 
Pupils  formerly  devoted  about  one-third  of  their  school 
time  to  the  study  of  arithmetic.  As  early  as  1850,  other 
subjects  which  were  demanding  admission  to  the  curric- 
ulum sought  to  curtail  the  time  given  to  arithmetic.  About 
1875  arithmetic  was  displaced  as  the  dominant  factor  in 
determining  promotion.  To-day  fifty  of  the  leading  cities 
of  the  United  States  devote  an  average  of  15.2  per  cent  of 


WASTE  IN  ARITHMETIC  121 

the  school  time  to  the  subject.  The  decline  in  the  time 
devoted  to  arithmetic  is  not  due  to  a  feeling  that  arith- 
metic is  less  worthy,  but  to  the  demands  of  other  subjects 
for  admission.  Arithmetic  will  probably  maintain  its  posi- 
tion of  prominence  for  many  years,  but  it  will  eventually 
surrender  a  portion  of  its  time  to  such  other  topics  as  are 
sufficiently  thought  out  and  systematized  as  to  justly  claim 
a  share.  A  sufficient  number  of  school  hours  are  still 
devoted  to  the  study  of  arithmetic.  What  the  schools  need 
is  not  more  time  for  the  subject,  but  a  clearer  realization 
of  the  purposes  for  which  arithmetic  should  be  taught  and 
a  better  adjustment  of  means  to  ends.  We  need  to  econo- 
mize the  time  that  is  now  devoted  to  the  subject  rather 
than  waste  it  by  aimless  and  unsystematic  instruction. 

i 

Applications  of  Arithmetic 

It  is  wasteful  to  teach  arithmetic  without  sufficiently 
emphasizing  the  applications  of  the  subject  to  life.  The 
study  should  help  the  pupil  to  interpret  life  from  the 
quantitative  point  of  view,  and  unless  it  does  this  the 
subject  is  not  fulfilling  its  full  purpose.  A  pupil  should 
be  taught  his  arithmetic  so  that  he  can  apply  it  to  prob- 
lems outside  of  the  book.  Too  many  pupils  can  solve  com- 
plicated problems  about  prisms,  pyramids,  and  cones,  but 
cannot  find  the  volume  of  their  father's  coal  bin  or  the 
amount  of  dirt  that  must  be  removed  in  digging  a  cellar. 
Unless  the  facts  of  the  classroom  are  interpreted  in  terms 
of  facts  outside  the  classroom  the  teaching  of  the  subject 
is  not  efficient.  There  should  be  a  direct  and  immediate 
relation  between  many  of  the  problems  of  the  arithmetic 
class  and  outside  experiences  that  demand  similar  knowl- 
edge. Many  of  the  experiences  of  the  pupil  in  thg  school 
should  vitalize  the  experiences  outside  of  the  school,  and 


122  HOW  TO  TEACH  AE1THMETIC 

many  of  the  experiences  outside  of  the  school  should  be 
utilised  to  clarify  the  work  of  the  school.  Pupils  are  too 
often  required  to  solve  problems  of  whose  use  and  applica- 
tion they  can  have  no  clear  conception. 

Unity  of  Arithmetic 

It  is  wasteful  to  teach  arithmetic  as  a  multitude  of  un- 
related topics  when  in  reality  the  subject  contains  only  a 
few  distinct  processes.  The  pupil  should  appreciate  the 
fact  that  the  same  process  may  appear  under  various 
phases  of  the  subject.  One  who  has  no  appreciation  of  the 
unity  and  the  simplicity  of  arithmetic  has  failed  to  grasp 
the  full  significance  of  the  subject;  arithmetic  to  him  is 
a  maze  of  rules  and  processes.  The  teacher  should  show 
the  pupil  how,  from  a  few  definitions  and  fundamental 
processes,  the  entire  science  is  developed  step  by  step.  To 
present  a  new  topic  without  showing  its  relation  to  those 
that  have  preceded  is  wasteful  teaching. 

The  Thought  Side  of  Arithmetic 

There  is  waste  in  the  teaching  of  arithmetic  because  not 
enough  attention  is  given  to  the  development  of  the  pupil's 
power  to  reason.  Not  enough  training  is  given  in  the 
selecting  of  the  proper  process  to  be  used  in  a  given 
situation. 

Mechanical  processes  are  of  great  importance  in  arith- 
metic, and  our  pupils  have  not  become  too  expert  in  them ; 
but  in  many  schools  mechanical  ability  has  been  developed 
largely  at  the  expense  of  the  ability  to  see  relations  and  to 
think,  to  select  appropriate  processes  with  certainty.  Fre- 
quently too  small  a  proportion  of  the  arithmetic  period  is 
given  to  developing  the  thought  side  of  the  subject.  The 
pupil  should  be  able  to  perform  the  fundamental  opera- 


WASTE  IN  ARITHMETIC  123 

tions  with  integers  and  fractions  with  reasonable  facility 
and  with  a  high  degree  of  accuracy;  but  it  is  necessary 
also  that  he  should  know  when  the  performance  of  a  given 
process  is  necessary  in  the  solution  of  a  given  problem.  If 
the  judgment  of  the  pupil  is  to  be  developed  in  arithmetic, 
the  teacher  should  exercise  great  care  that  he  does  not 
hamper  it  by  the  imposition  of  numerous  rules.  Much  of 
mechanical  teaching  is  wasteful  teaching. 

Assignments 

There  is  much  waste  of  time  and  of  effort  in  the  study 
of  arithmetic  because  the  teacher  does  not  make  the  assign- 
ment with  definiteness  and  precision,  and  as  a  result  of  this 
the  study  hour  is  one  of  blind  grouping  and  discouraging 
failure,  resulting  in  disgust  for  the  study  and  in  dawdling 
habits.  All  assignments  are  intended  to  afford  growth 
toward  independence  and  initiative  in  thought  and  action. 
The  immediate  purpose  of  an  assignment  varies,  but  in 
general  its  purpose  is  to  guide  the  pupil  in  preparing  for 
whatever  the  new  recitation  will  present,  to  direct  him 
toward  the  accomplishment  of  what  he  can  do  by  himself, 
and  to  save  time  in  the  recitation  proper. 

The  teacher  should  always  have  a  definite  and  satisfac- 
tory reason  for  assigning  each  lesson.  This  reason  should 
be  based  on  the  teacher's  knowledge  of  the  pupils,  their 
interest  and  needs,  and  on  a  knowledge  of  the  lesson.  A 
well-made  assignment  saves  time  in  succeeding  recitations 
and  makes  the  recitation  more  profitable  than  it  could  pos- 
sibly be  without  the  preparation  of  the  child's  mind  that 
is  afforded  through  the  assignment. 

In  the  lower  grades  especially,  all  assignments  should  be 
made  with  great  care  and  precision.  As  the  pupils  grow 
in  power  they  may  sometimes  be  permitted  to  suggest 


124  HOW  TO  TEACH  AEITHMETIC 

the  assignment,  which  will  be  discussed  and  amended  by 
the  teachers.  Some  classes,  late  in  the  grades,  may  occa- 
sionally be  trusted  to  make  the  assignment. '  This  affords 
an  excellent  means  of  strengthening  the  selective  power 
and  of  encouraging  originality. 

A  good  assignment  will  put  the  pupil  in  a  mood  to  work 
on  his  lesson,  and  will  stimulate  him  to  attack  and  solve 
the  problems  before  him. 

Explanations 

It  is  questionable  how  much  explanation  should  be  given 
by  the  teacher  when  the  assignment  is  made.  Some  assert 
that  the  teacher  should  not  point  out  the  difficulties,  but 
should  leave  the  child  free  and  unhampered  by  any  assist- 
ance. This  view  will  probably  entail  a  great  waste  of  time 
and  energy  in  most  classes.  It  often  leads  to  the  magnify- 
ing of  trivialities,  and  results  in  unorganized  knowledge. 

It  is  probably  good  practice  in  the  lower  grades  for  the 
teacher  to  explain  nearly  all  the  difficulties,  and  in  the 
upper  grades  to  make  the  assignment  in  such  a  way  as  to 
assist  in  removing  them.  One  measure  of  the  value  of  an 
assignment  is  the  degree  of  interest  aroused  and  main- 
tained in  the  class  in  its  study  apart  from  the  teacher.  If 
the  teacher  knows  both  his  pupils  and  the  subject  to  be 
taught,  he  should  be  able  to  arouse  the  interest  of  the 
pupils  in  the  subject. 

If  in  the  recitation,  or  even  when  making  the  assign- 
ment, the  teacher  works  up  to  an  absorbing  point  and  then 
leaves  it  in  suspense,  the  pupils  will  return  to  the  subject 
with  keen  interest  during  the  study  hour. 

Just  enough  time  should  be  devoted  to  the  assignment 
to  make  it  clear  and  definite — to  remove  the  obscurities 
and  certain  difficulties,  and  to  create  enough  interest  to 
insure  further  study.  Sometimes  this  can  be  done  in  a  few 


WASTE  IN  ARITHMETIC  125 

minutes ;  at  other  times  it  will  require  most  of  the  recita- 
tion period.  The  assignment  is  a  place  for  raising  prob- 
lems and  creating  interest  in  those  problems.  Haste  and 
slovenliness  are  to  be  avoided,  because  they  are  destructive 
to  energetic  effort  on  the  part  of  the  pupils.  No  general 
directions  can  be  given  as  to  the  proper  time  to  make  an 
advance  assignment.  Usually  it  should  be  made  at  the 
beginning  of  the  recitation  period.  If  the  assigning  of  the 
advance  lesson  is  postponed  until  the  close  of  the  period, 
teachers  frequently  find  that  they  have  not  allowed  them- 
selves sufficient  time,  and  the  result  is  a  hurried  assign- 
ment, lacking  both  definiteness  and  precision. 

A  teacher  has  no  right  to  expect  a  pupil  to  prepare  with 
precision  an  assignment  that  was  not  made  with  care  and 
with  definiteness.  It  sometimes  happens  that  the  a'dvance 
assignment  is  to  depend  largely  upon  points  to  be  devel- 
oped during  the  recitation  period  or  upon  the  extent  to 
which  the  recitation  has  been  satisfactory.  Under  such 
conditions  the  teacher  cannot  know  the  appropriate  length 
for  the  assignment  or  what  features  should  be  made  most 
prominent  until  late  in  the  period.  Some  excellent  teach- 
ers always  make  the  advance  assignment  at  the  close  of  the 
period,  but  the  custom  is  not  one  that  can  be  generally 
practiced  with  the  best  results.  Individual  assignments, 
such  as  telling  a  pupil  that  he  will  be  held  responsible  for 
the  solution  and  explanation  of  a  given  problem  at  the 
next  recitation,  may  properly  be  made  in  the  midst  of  the 
recitation. 

Methods  of  Study 

One  of  the  greatest  sources  of  waste  in  education  is 
found  in  improper  methods  of  study.  Many  earnest  pupils 
who  are  anxious  to  study  do  not  know  how  to  do  so  "inde- 
pendently, intelligently,  or  economically. "  The  teacher 


126  HOW  TO  TEACH  ARITHMETIC 

should  do  all  that  he  can  to  assist  the  pupil  to  form  good 
habits  of  study.  Two  books  which  discuss  the  general 
problem  of  how  to  study  should  be  noted,  Sandwick's 
"How  to  Study  and  What  to  Study "  and  McMurray's 
' '  How  to  Study. ' '  The  most  extensive  investigation  of  the 
various  methods  of  teaching  pupils  how  to  study  mathe- 
matics is  to  be  found  in  Part  I  of  the  Thirteenth  Year 
Book  of  the  National  Society  for  the  Study  of  Education. 
These  books,  especially,  contain  numerous  suggestions  on 
the  general  problem,  and  some  of  them  apply  to  the  study 
of  arithmetic. 

Pupils  should  learn  to  overcome  difficulties  without  the 
aid  of  the  teacher.  A  spirit  of  self-reliance  and  of  tenacity 
of  purpose  are  excellent  assets  for  any  pupil.  Interest  in 
one 's  work  may  be  initiated  and  augmented  by  consistent 
application  to  a  task  that  may  originally  have  been  without 
interest.  Many  a  pupil  has  become  tremendously  interested 
in  his  mathematics  by  persevering  over  tasks  that  were  at 
first  uninteresting. 

If  a  pupil  is  to  prepare  his  mathematics  assignment  with 
economy  of  time  and  of  effort,  the  conditions  under  which 
he  studies  should  be  such  as  to  admit  of  a  high  degree  of 
concentration  upon  the  work  in  hand.  The  habit  of  con- 
centrating the  attention  is  of  great  value  in  studying  any 
subject,  and  especially  mathematics.  Many  pupils  do  not 
seem  to  realize  the  necessity  of  selecting,  when  this  is  pos- 
sible, an  environment  that  is  conducive  to  study,  and  that 
will  permit  concentration  upon  the  task.  If  parents  and 
pupils  can  be  convinced  that  under  proper  conditions  twice 
the  work  may  be  accomplished  in  half  the  time,  a  great 
gain  has  been  made. 

Waste  in  Mechanical  Procedure 

There  is  waste  of  the  recitation  period  because  the 
teacher  permits  pupils  to  form  dawdling  habits  of  work, 


WASTE  IN  AKITHMETIC  127 

especially  at  the  blackboard.  This  can  usually  be  avoided 
by  attention  on  the  part  of  the  teacher.  In  many  classes 
much  time  is  wasted  in  such  procedures  as  the  calling  of 
the  class  roll,  passing  to  the  blackboard,  and  the  distribu- 
tion of  papers.  The  writers  visited  a  class  in  arithmetic  in 
which  the  teacher  consumed  eight  of  the  forty  minutes 
allotted  for  the  recitation  in  the  calling  of  the  roll  and  the 
distribution  of  papers.  During  this  time  no  comments  or 
suggestions  were  made  in  regard  to  any  of  the  work.  This 
teacher  wasted  about  20  per  cent  of  the  recitation  period  in 
doing  things  that  should  have  been  done  with  less  confu- 
sion in  one-tenth  of  the  time.  The  recitation  that  followed 
reflected  the  dawdling  habits  and  unsystematic  procedures 
of  the  teacher.  Subsequent  observations  revealed  the  fact 
that  the  teacher  wasted  at  least  20  per  cent  of  the  recita- 
tion period  each  day  in  such  ways  as  have  been  indicated. 
As  the  teacher  is,  the  class  is.  A  teacher  who  does  not  work 
in  class  with  economy  of  time  and  of  effort  will  soon  find 
that  many  of  the  pupils  have  developed  habits  of  work 
quite  similar  to  his  own  in  that  subject.  If  a  teacher  wishes 
the  maximum  amount  of  work  to  be  accomplished  during 
the  recitation  period,  even  the  minute  details  of  the  work 
must  be  planned  with  this  end  in  view.  Not  every  pupil 
in  a  class  will  adjust  himself  to  the  ideals  of  the  teacher, 
but  it  is  certain  that  unless  the  teacher  himself  has  proper 
standards  of  work  the  general  average  of  the  class  in  this 
respect  will  be  materially  lowered. 

Reviews  and  Examinations 

Some  educators  would  eliminate  reviews  and  examina- 
tions from  school  work.  They  say  that  such  eliminations 
will  delight  the  American  boy  and  that  no  compulsion  will 
be  necessary  to  induce  him  to  attend  school.  School  life 


128  HOW  TO  TEACH  AEITHMETIC 

will  be  a  continuous  round  of  pleasure,  and  there  will  be 
no  work  to  make  Jack  a  dull  boy.  Such  radical  proposals 
are  without  social  or  psychological  warrant.  Sociology  in- 
forms us  that  it  is  the  part  of  wisdom  to  hold  onto  those 
things  that  long  and  successful  experience  have  proved  to 
be  worth  while.  Psychology  informs  us  that  drills  and 
reviews  are  necessary  in  the  fixing  of  habits  and  in 
organizing  material. 

It  is  a  fundamental  principle  of  psychology  that  if  one 
is  equally  hazy  about  two  things  formerly  learned,  one 
months  ago  and  the  other  recently,  it  requires  a  smaller 
amount  of  effort  to  recall  and  fix  the  older  of  the  two  than 
it  does  to  recall  and  fix  the  one  more  recently  acquired. 

Eeviews  properly  distributed  pay  tremendous  dividends 
in  the  economy  of  mental  life.  Surely  the  .old  practice, 
which  provided  for  the  automatic  mastery  of  the  funda- 
mentals, should  not  be  discontinued  without  a  hearing. 
Many  things  still  need  to  be  reduced  to  the  automatic  or  to 
be  firmly  fixed  in  the  mind  by  means  of  drills  and  reviews. 
No  true  scholastic  attainment  or  efficient  education  is  pos- 
sible without  a  reasonable  use  of  reviews  and  drills. 

Our  pupils  to-day  are  in  more  danger  of  suffering  from 
intellectual  starvation  than  from  mental  dyspepsia.  A 
school  is  organized  and  maintained  in  order  that  it  may  be 
instructed,  and  reviews  are  an  essential  part  of  efficient 
instruction.  The  social  or  educational  reformer  who  would 
eliminate  all  drill  and  all  reviews  is  striking  at  one  of  the 
essentials  of  instruction. 

In  individual  lessons  attention  is  focused  primarily  upon 
the  various  parts  of  the  subject-matter  rather  than  upon 
the  unity  of  the  parts.  A  good  review  transfers  the  atten- 
tion to  the  larger  relations.  The  unity  of  a  subject  is 
likely  to  be  lost  in  the  multitude  of  detail,  and  a  compre- 
hensive and  thorough  review  is  necessary,  not  only  to  fix 


WASTE  IN  ARITHMETIC  129 

the  important  points  more  firmly  in  the  mind,  but  to 
produce  that  thoroughly  organized  and  well-articulated 
knowledge  which  is  such  a  valuable  asset.  Reviews  should 
appeal  to  both  the  eye  and  the  ear. 

The  ingenuity  and  skill  of  the  teacher  are  frequently 
taxed  to  the  utmost  in  order  to  devise  ways  and  means  of 
instilling  a  high  degree  of  interest  into  a  review  lesson.  .It 
is  not  necessary  that  the  subject-matter  and  the  method  of 
presentation  be  the  same  as  when  the  topics  were  previously 
studied.  Indeed,  the  advance  that  has  been  made  will 
frequently  suggest  a  broader  view  and  a  better  method. 

Thorough  and  comprehensive  reviews  are  especially 
necessary  at  the  close  of  the  year's  work.  Investigations 
on  this  point  indicate  that  such  reviews  are  economical  in 
that  they  save  time  in  the  work  of  the  succeeding  year.  A 
brief  review  of  the  principal  features  of  the  previous 
year's  work  should  also  be  given  during  the  first  week  or 
two  of  the  school  year. 

It  is  the  custom  to-day  in  some  localities  to  decry  exami- 
nations and  to  urge  that  they  be  eliminated  from  the  school. 
It  is  doubtless  true  that  examinations  in  the  past  have 
frequently  been  given  with  little  or  no  though!:  as  to  their 
real  purposes. 

Examinations  should  usually  be  comprehensive  in  char- 
acter :  their  purpose  should  be  to  aid  the  pupils  in  organiz- 
ing and  correlating  the  knowledge  acquired  into  a  coherent 
and  systematic  unit.  The  examination  should  be  regarded 
as  a  means  to  an  end,  and  not  as  an  end  in  itself.  Too 
often  the  examination  is  regarded  simply  as  a  means  of 
testing  the  memory  for  specific  and  unrelated  facts.  It  is 
proper  that  a  part  of  an  examination  should  be  devoted  to 
this  purpose,  but  the  important  facts  to  be  tested  should 
be  the  ability  to  organize  and  to  apply. 

Examinations  given  by  the  teacher,  if  well  planned  and 


130  HOW  TO  TEACH  AKITHMETIC 

wisely  constructed,  frequently  aid  the  teacher  to  discover 
individual  or  class  weaknesses,  and  to  furnish  a  basis  for 
further  instruction.  Such  examinations,  if  given  with  due 
regard  for  the  welfare  of  the  pupils,  are  justifiable. 

Examinations  given  by  superintendents,  principals,  or 
supervisors  usually  have  as  their  aim  the  setting  of  stand- 
ards of  work  throughout  the  system,  the  testing  of  the 
efficiency  of  the  course  of  study,  and  the  discovery  of  indi- 
vidual weakness.  Such  examinations,  if  properly  con- 
ducted, may  be  made  an  effective  administrative  device. 
Broad-minded  teachers  usually  welcome  the  opportunity  to 
have  their  work  tested. 

The  present  tendency  is  to  decrease  the  frequency  of 
examinations  and  to  use  more  care  in  their  preparation. 
The  marks  received  on  examinations  formerly,  determined 
whether  a  pupil  was  to  be  promoted.  This  is  not  true  in 
many  schools  to-day.  The  practice  of  giving  "finals"  is 
now  uncommon.  The  memorites  type  of  examination  is 
being  replaced  by  those  which  require  old  knowledge  to  be 
utilized  not  in  isolation  but  in  new  situations.  More  ques- 
tions demanding  thought,  judgment,  and  choice  are  given 
to-day  than  in  former  years. 


PART  THREE 

CHAPTER,  X 
PEIMARY  ARITHMETIC 

Preliminary  Statement 

The  arithmetic  which  children  are  first  put  to  work  upon 
should  be  closely  related  to  their  lives.  Arithmetic  is  one 
of  the  agencies  consciously  prescribed  by  society  for  giving 
children  control  over  a  particular  phase  of  their  environ- 
ment. This  control,  however,  cannot  be  economically  or 
advantageously  acquired  unless  children  are  provided  with 
normal  situations  that  provoke  natural  reactions.  A 
genuine  mastery  of  number  in  the  early  school  years  is 
gained  by  using  it  in  a  concrete  manner — in  the  construc- 
tion of  play-houses,  in  games,  in  weighing,  measuring,  and 
counting  those  objects  and  relationships  that  represent  the 
daily  enterprises  of  child  life. 

Much  of  the  teaching  of  arithmetic  has  not  been  domi- 
nated by  this  ideal.  In  the  past,  instruction  was  charac- 
terized by  the  memorizing  of  rules,  formulas,  and  exam- 
ples. Little  attempt  was  made  to  relate  the  materials  and 
methods  of  arithmetic  to  social  and  industrial  life.  The 
advent  of  a  multitude  of  new  subjects,  such  as  nature 
study,  gardening,  manual  training,  domestic  economy,  and 
agriculture,  had  much  to  do  in  converting  the  old  reflective 
school  into  a  more  active  school.  The  presence  of  studies 
in  the  curriculum  is  no  longer  justified  by  their  hypothet- 
ical mind-training  value.  Now  they  are  justified  by  the 
number  of  relations  of  identity  they  have  with  the  world. 

131 


132  HOW  TO  TEACH  AEITHMETIG 

This  influence  has  spread  across  and  modified  instruction 
in  the  older  subjects.  The  new  movement  is  seen  in  arith- 
metic in  the  elimination  of  materials  no  longer  socially 
serviceable,  in  the  addition  of  materials  closely  related  to 
the  business  practice  of  current  life,  and  in  the  rationaliz- 
ing of  instruction  through  object  work. 

Nature  of  Primary  Arithmetic 

Primary  arithmetic  has  been  restricted  by  common  con- 
sent to  a  knowledge  and  mastery  of  the  fundamental 
operations  as  expressed  in  integers  and  in  fractions.  It  is 
not  vocational  in  the  sense  that  its  processes  are  named 
after  particular  occupations.  This  division  of  the  subject- 
matter  is  due  largely  to  the  preparatory  character  and 
value  of  these  fundamental  processes.  They  are  the  intel- 
lectual tools  which  all  must  use  in  their  later  life,  no 
matter  what  occupation  they  choose.  Reform  tendencies 
in  method  have  reconstructed  the  traditional  presentation 
of  these  tools  by  rules  and  formulas  to  the  more  rational 
discovery  of  their  nature  in  concrete  problems. 

Dominance  of  Methods  in  the  Teaching  of  Arithmetic 

Mathematical  instruction  was  long  dominated  by  the 
logical  theories  and  forms  of  organization  of  men  of 
science.  Each  fact  was  taught  to  show  its  relation  to 
other  facts  of  the  subject,  and  not  to  connect  it  with  some 
common  operations  of  life.  Courses  of  study  were  con- 
structed in  the  office  of  the  superintendent  and  handed 
over  ready-made  to  the  classroom  teacher.  Those  special 
and  personal  modifications  so  essential  in  reconstructing 
the  experiences  of  children  were  seldom  encouraged.  For 
this  reason  teachers  taught  as  they  had  been  taught  by 
men  of  science,  or  as  they  had  been  told  to  teach,  and  not 
as  the  exigencies  of  the  situation  demanded.  Instruction 


PRIMARY  ARITHMETIC  133 

was  logical,  not  psychological.  The  measure  of  the  value 
of  a  fact  was  its  relation  to  the  other  facts  in  an  organized 
scheme,  and  not  its  recurrence  in  daily  life. 

Logical  vs.  Psychological  Method 

The  logical  method  or  arrangement  of  material  repre- 
sents the^  adult  or  the  scientific  point  of  view,  while  the 
psychological  method  or  arrangement  represents  the  man- 
ner in  which  children  normally  approach  and  master  the 
situations   of   a   subject.     When   materials   are   logically 
arranged  there  is  always  some  central,   organizing  inte- 
grating principle.  When  they  are  psychologically  arranged 
there  is  a  distinct  attempt  to  present  materials  to  harmonize 
with  the  psychological  conditions  of  childhood.    The  spiral 
or  concentric  circle  plan  is  built  upon  the  theory  that 
instruction  should  proceed  from  the  simple  to  the  less 
simple,  and  from  the  less  simple  to  the  more  complex.    The 
facts  and  skills  first  presented,  whether  they  are  addition, 
subtraction,  multiplication,  or  division  recur  later  in  more 
complicated  form,  and  reappear  again  and  again  as  the 
circles  widen.     Materials  are  thus  adjusted  to   suit  the 
various  psychological  stages  or  maturity  levels  of  children. 
These  methods  have  had  a  reactionary  influence  upon 
each  other.    Naturally  the  older  one  is  the  logical.    It  is  so 
ingrained    in    our    practice    that    teaching    has    suffered 
from  its  baneful  effects.    The  spiral  plan,  like  every  reform 
movement,  tended  to  swing  practice  to  an  unnatural  ex- 
treme.    Compromise  was  inevitable.     Now  some  phases  of 
arithmetic,  for  example,  parts  of  denominate  numbers  are 
taught  topically,  while  the  fundamentals,  whether  integers 
or  fractions,  are  taught  spirally. 

Formal  Discipline 

This  scientific  justification  for  mathematical  instruction, 
which  came  down  from  the  university  and  took  root  in  the 


134  HOW  TO  TEACH  AEITHMETIG 

primary  schools,  was  supplanted  by  the  doctrine  of  formal 
discipline,  which  invited  support  and  justified  both  the 
logical  form  of  organization  and  the  presence  of  old  ma- 
terials in  the  curriculum  on  the  ground  that  they  trained 
the  mind  and  hence  were  of  value  in  later  life,  no  matter 
how  remotely  they  were  related  to  the  processes  of  daily 
life.  This  doctrine  afforded  a  sanction  for  the  retention  of 
obsolete  materials  and  antiquated  methods.  Although  more 
people  were  interested  in  building  and  loan  associations  and 
in  insurance  than  in  the  greatest  common  divisor,  cube  root, 
or  partnership,  the  teacher  argued  vigorously  for  the  reten- 
tion of  the  latter  on  the  ground  of  their  mind  training 
value. 

The  disciplinary  defense  gradually  began  to  break  down 
before  the  demands  of  a  commercially  prosperous  public. 
A  commercial  age  is  always  a  period  of  great  educational 
transition  and  advance.  It  seeks  to  justify  the  practices 
of  the  school  in  terms  of  needs  and  conditions  outside  of 
school.  Business  men  exalted  instruction  in  those  par- 
ticular arithmetical  processes  that  were  needed  in  business 
practice,  and  discouraged  others.  Business  utility  op- 
erated both  as  an  eliminating  and  as  a  selective  agency. 
Impractical  and  obsolete  materials  were  discarded  and 
new  materials  were  introduced.  The  restricted  point  of 
view  of  the  business  man  is  now  being  transformed  by  a 
wider  social  point  of  view  which  calls  for  that  instruction 
that  conserves  our  common  human  obligations  rather  than 
trains  for  a  specific  vocation. 

Time  to  Introduce  Arithmetic 

Opinion  differs  as  to  the  proper  time  for  beginning  the 
study  of  arithmetic.  In  some  schools  a  definite  part  of  the 
school  day  is  set  apart  during  the  first  school  year  for  the 
study  of  numbers.  In  other  schools  such  work  is  post- 


PRIMARY  ARITHMETIC  135 

poned  until  the  second  school  year,  and  in  a  few  schools  no 
formal  study  of  numbers  is  taken  up  until  the  third  school 
year.  In  the  more  advanced  European  countries  arith- 
metic is  studied  during  the  first  school  year  and  the  ten- 
dency seems  to  be  in  that  direction  in  this  country.  Until 
about  one  hundred  years  ago  arithmetic  was  never  taught 
to  pupils  just  entering  school.  Many  think  that  any  regu- 
lar and  systematic  attention  to  number  is  premature  in 
the  first  school  year  and  that  the  time  should  be  spent  in 
widening  the  pupil's  activities  and  knowledge  through  ele- 
mentary nature  study,  garden  making,  games,  drawing, 
and  constructive  exercises.  They  contend  that  it  is  legiti- 
mate to  introduce  only  so  much  of  number  work  as  will 
serve  a  definite  end  in  these  subjects.  The  work  in  arith- 
metic, they  maintain,  should  be  incidental  and  not  formal 
and  systematic  in  the  first  school  year.  Whenever  a  quan- 
titative relationship  presents  itself  in  any  school  activity 
they  would  attempt  to  make  it  clear.  They  maintain  that 
such  a  plan  tends  to  develop  number  ideas  naturally  and 
does  not  force  them  and  hence  makes  the  subject  more 
attractive  to  the  child. 

Pupils  are  asked  to  note  the  number  in  the  class,  the 
number  absent  or  tardy ;  to  compute  the  number  of  pencils, 
or  books  needed  for  the  class  or  for  a  given  row.  Games 
offer  numerous  opportunities  for  introducing  number  ideas 
because  of  the  necessity  of  keeping  score,  etc. 

Others  contend  that  when  a  pupil  comes  to  school  he  is 
just  as  anxious  to  learn  the  fundamental  relation  of  num- 
bers as  to  learn  to  read,  and  that  it  is  as  unwise  to  postpone 
one  as  the  other.  Too  often  incidental  teaching  becomes 
accidental  and  perfunctory  teaching,  and  a  pupil  finishes 
his  first  school  year  with  his  number  ideas  but  little  ad- 
vanced. It  is  wise  to  motive  the  number  work  in  the  first 
and  in  all  other  grades,  but  many  teachers  wrongly  assume 


136  HOW  TO  TEACH  AEITHMETIO 

that  interest  in  a  subject  for  its  own  sake  is  a  motive  that 
should  not  be  considered.  Most  young  children  naturally 
like  arithmetic,  but  many  of  them  soon  come  to  dislike  it 
simply  because  of  poor  teaching  in  the  lower  grades.  One 
of  the  great  wastes  in  arithmetic  lies  just  here.  The  inter- 
est of  many  pupils  in  the  study  of  numbers  is  deadened 
and  as  interest  wanes  accomplishment  fails  and  the  pupil 
finds  that  he  has  developed  a  positive  dislike  for  the  sub- 
ject. In  general,  the  methods  of  teaching  in  the  primary 
grades  are  more  carefully  thought  out  and  systematized 
than  in  the  intermediate  grades,  but  a  great  responsibility 
rests  upon  the  teachers  of  the  primary  grades  for  giving  a 
pupil  as  good  a  start  as  is  possible  for  him  in  the  subject. 

Correlations 

Innumerable  opportunities  arise  in  other  subjects  to 
teach  number.  The  objects  used  should  be  those  in  which 
the  children  are  interested.  Much  comparatively  dull  work 
has  bee,n  done  in  the  past  by  using  objects  having  had  only 
an  adventitious  interest.  Highly  decorated  cards,  curiously 
carved  animals  and  queer  looking  sticks  do  not  serve  the, 
purpose  best.  The  objects  used  should  be  changed  fre- 
quently. It  is  stupid  to  teach  all  the  number  operations 
by  using  beans. 

Children  must  learn  to  find  their  seats,  their  place  in  the 
line,  to  get  six  erasers,  three  pieces  of  crayon,  to  distribute 
five  pencils  or  four  cards,  and  to  turn  to  a  given  page. 
Opportunities  of  this  kind  for  the  teaching  of  counting 
will  not  be  overlooked  by  the  resourceful  teacher. 

Correlation  With  Constructive  Work 

Every  lesson  in  constructive  work  affords  many  oppor- 
tunities for  teaching  numbers.  Every  lesson  upon  the  tri- 
angle, square  or  rectangle  is  to  some  extent  a  lesson  in 


PRIMAEY  ARITHMETIC  137 

numbers.  When  such  materials  are  used  the  teacher  should 
make  no  attempt  to  differentiate  the  four  fundamental 
operations. 

A  multitude  of  interesting  constructive  devices  have 
been  invented  for  teaching  number,  each  of  which  is  of 
value  not  only  as  an  occupational  device,  but  because  of 
the  intellectual  by-products  resulting  from  its  mastery. 
Among  other  occupational  devices  listed  for  the  first  grade 
are :  square  seed  box,  seed  envelopes,  table,  the  three  chairs 
of  the  three  bears,  the  three  beds,  basket,  sled,  soldier  cap, 
boat,  folding  basket,  square  prism,  cube,  triangular  prism, 
pyramid,  closed  seed  box,  trunk,  cradle,  bath  tub,  candy 
box,  match  stand.1  Most  of  the  above  exercises  can  be 
taught  without  using  the  ruler,  but  they  cannot  be  taught 
without  training  the  number  sense  of  the  children.  As  soon 
as  the  pupils  have  acquired  some  skill  in  handling  the 
ruler  the  number  of  objects  they  can  make  is  increased. 
Some  of  those  described  in  Mr.  Worst's  book  are:  paper 
chairs,  paste  trays,  postage  stamp  holder,  cornucopia,  mat, 
woven  basket,  thread  winder,  color  exercises,  pin  holder. 

The  Use  of  Objects 

Successful  primary  work  to-day  is  almost  intuitively 
associated  with  object  teaching.  No  good  primary  teacher 
would  think  of  attempting  to  teach  numbers  without  a  sup- 
ply of  objects.  If  these  are  judiciously  selected  they  serve 
as  the  school's  best  substitute  for  those  outside  situations 
that  attract  the  native  tendencies  of  children.  At  any 
rate  they  are  the  school 's  one  best  means  of  making  concrete 
the  abstract  numerical  concepts  that  up  to  the  middle  of 

i  The  steps  used  in  making  these  objects  are  described  in  a  book 
entitled  "Constructive  Work/'  by  Edward  F.  Worst,  Superintendent 
of  Schools,  Joliet,  Illinois,  published  by  A.  W.  Mumford,  Chicago, 
Illinois. 


138  HOW  TO  TEACH  ABITHMETIC 

the  nineteenth  century  were  taught  by  purely  memoriter 
methods.  Pestalozzi  has  been  called  the  father  of  object 
teaching.  This  is  correct  only  in  the  sense  that  he  restored 
the  use  of  the  object  to  the  schools.  As  children  gain  in 
power  of  independent  thought,  that  is,  in  ability  to  or- 
ganize and  condense  their  experiences,  the  number  of  ob- 
jects used  in  teaching  decreases;  but  they  never  entirely 
disappear,  for  they  are  always  of  value  in  comprehending 
and  interpreting  new  situations. 

Dr.  Suzzallo's  discussion  of  the  materials  of  objective 
teaching  is  particularly  illuminating.  His  first  indictment 
is  the  artificiality  of  the  materials  employed.  "  Primary 
children  count,  add,  etc.,  with  things  they  will  never  be 
concerned  with  in  life.  Lentils,  sticks,  tablets  and  the  like 
are  the  stock  objective  stuff  of  the  schools,  and  to  a  con- 
siderable degree  this  will  always  be  the  case.  Cheap  and 
convenient  material  suitable  for  individual  manipulation 
is  not  plentiful.  But  instances  where  better  and  more  nor- 
mal materials  have  been  used  are  frequent  enough  in  the 
best  schools,  to  warrant  the  belief  that  more  could  be  done 
in  this  direction  in  the  average  classroom.  The  '  playing 
at  store,'  and  the  use  of  actual  applications  of  the  tables 
of  weights  and  measures  are  cases  that  might  be  cited." 

His  second  indictment  is  that  the  materials  have  too 
restricted  a  range.  The  range  of  materials  is  almost  di- 
rectly dependent  upon  the  resourcefulness  of  the  teaching 
staff.  "When  teachers  and  supervisors  know  what  they 
want  and  need  in  teaching,  there  will  be  comparatively 
little  difficulty  in  inducing  beards  of  education  to  buy  it. 

His  third  indictment  is  the  limited  use  of  the  materials 
on  hand.  "It  is  too  frequently  the  case  that  the  teacher 
will  treat  the  fundamental  addition  combinations  with  one 

NOTE — For  an  elaborate  discussion  of  "The  Teaching  of  Primary 
Arithmetic/'  see  Teachers'  College  Record,  March,  1911,  Suzzallo. 


PRIMARY  ARITHMETIC  139 

set  of  objects,  e.  g.,  lentils.  In  all  the  child's  objective 
experience  within  that  field  there  are  two  persistent  asso- 
ciations— ' lentils'  and  the  relations  of  addition.  A  wide 
variation  in  the  objective  material  used  would  make  teach- 
ing more  effective,  particularly  with  young  children. " 

His  fourth  indictment  is  that  the  materials  of  objective 
teaching  have  been  too  narrowly  interpreted.  He  advocates 
a  more  extensive  use  of  pictures,  both  in  text-books  and  in 
teaching,  and  of  geometric  figures  and  diagrams  which  are 
of  value,  though  in  a  more  restricted  way,  in  extending 
the  range  of  concrete  experiences. 

His  final  indictment  is  that  unsympathetic  and  conserva- 
tive teachers  use  objects  in  a  highly  artificial  and  disor- 
ganized way.  On  the  other  hand  the  resourceful  and  pro- 
gressive teacher  secures  some  unity  even  in  the  materials 
she  handles;  she  uses  a  game,  some  community  activity, 
some  childish  interest,  permits  the  children  to  play  "store," 
or  conductor  on  a  street  car,  etc. 

The  wasteful  effects  of  undue  emphasis  upon  unrelated 
objects  may  be  shown  by  a  description  of  an  actual  recita- 
tion. The  number  fact  to  be  taught  was  6  +  3.  The  chil- 
dren were  led  to  see  that 

6  blocks  +  3  blocks  =  9  blocks. 

This  was  arrived  at  by  putting  6  blocks  in  one  group  and 
3  blocks  in  another,  combining  these  groups  and  then  count- 
ing, after  which  the  pupils  in  turn  repeated  the  "story." 
The  process  was  then  repeated  with  marbles  and  the  chil- 
dren told  the  "story," 

6  marbles  +  3  marbles  =  9  marbles. 

Then  followed  beans,  toothpicks,  pennies,  shoe-pegs,  peas, 
apples,  crayons,  pencils,  rulers.  Then  pictures  of  cherries, 


140  HOW  TO  TEACH  ARITHMETIC 

oranges,  nails,  oblongs,  square,  and  chairs  were  used.  Fol- 
lowing this  they  passed  to  remember  objects  and  repeated 
the  appropriate  " story"  for  adding  birds,  trees,  boys, 
lemons,  horses,  potatoes,  and  bricks. 

In  this  particular  school  all  of  the  primary  lessons  were 
taught  in  this  way  throughout  the  entire  year.  It  took  a 
year — a  whole  year — to  teach  the  number  facts  and  rela- 
tions from  1  to  10.  Had  these  children  been  ignorant  of 
these  particular  facts  a  few  illustrations  would  have  sup- 
plied them  with  an  adequate  basis  for  making  the  proper 
generalizations. 

Use  of  Games 

The  principal  device  currently  used  by  teachers  in  fix- 
ing the  fundamental  arithmetical  concepts,  is  the  group 
play  of  children.  Games  permit  of  group  work  where  the 
attention  is  predominatingly  spontaneous.  Consequently 
it  is  not  uncommon  to  find  children  mastering  the  number 
operations  under  the  free  and  wholesome  atmosphere  of 
plays.  The  fascination  of  the  game,  which  is  due  partly 
to  the  element  of  chance,  reduces  the  symptoms  of  fatigue 
and  supplies  a  motive  and  zest  for  the  work  of  the  fol- 
lowing day.  The  otherwise  artificial  recitation  of  the  ex- 
amination sort  is  thus  transformed  into  a  source  of  pleas- 
ure. One  of  the  practical  benefits  early  realized  is  that 
children  soon  become  skillful  enough  to  play  such  games 
independently  of  the  teacher. 

List  of  Games 

1.  Dominoes  5.  Card  games 

2.  Parchesi  6.  Flinch 

3.  Crokinole  7.  Hop  scotch 

4.  Bean  bag  8.  Ten  pins 


PKIMAEY  ARITHMETIC  141 

9.  Tag  12.  Lotto 

10.  Odd  and  Even  13.  Jackstones 

11.  Marbles 

Around  the  Circle.  The  digits  are  arranged  in  the  form 
of  a  clock  face  upon  the  blackboard.  Any  number  of  these 
digits  may  be  used.  A  digit  is  then  placed  in  the  center 
and  the  numbers  are  multiplied  by  it  as  rapidly  as  pos- 
sible. If  a  pointer  can  be  made  to  swing  on  a  pivot,  the 
game  may  be  varied  by  taking  the  numbers  on  which  the 
pointer  rests.  The  game  can  also  be  varied  so  as  to  relate 
to  addition  and  subtraction. 

Bird  Catcher.  A  variation  of  this  well-known  game  is 
to  have  children  sit  in  a  circle,  each  taking  a  number.  The 
pupil  in  the  center  gives  easy  examples.  When  the  result 
is  the  number  of  any  pupil  in  the  circle  that  pupil  holds 
up  his  hands.  When  a  number  already  agreed  upon  is  the 
result,  all  hold  up  their  hands. 

Buzz.  The  members  of  the  class  count  in  turn.  When 
a  given  number  or  any  of  its  multiples  is  reached,  they 
say  "buzz"  instead  of  number.  Those  pupils  fall  out  who 
miss  or  who  hesitate  in  their  answers. 

Climb  the  Ladder.  Draw  a  ladder  on  the  blackboard. 
Put  a  combination  on  each  step,  letting  each-  pupil  climb 
until  he  falls. 

Hide  and  Seek.  Combinations  are  placed  on  the  black- 
board with  the  different  parts  missing — as,  14  +  3  =  ( ? )  ; 
14+(?)=17;  (?)+3  =  17;  4x(*)=24.  In  getting  the  an- 
swer the  pupils  unconsciously  repeat  the  combination  and 
form  the  desired  habit.  This  is  a  very  desirable  form  of 
play,  and  it  is  characteristic  of  the  kind  of  game  that  a 
teacher  can  make  for  herself.  Instead  of  the  symbol  (?) 
it  is  often  advantageous  to  use  X,  thus  giving  a  valuable 
algebraic  form  that  will  be  helpful  at  a  later  time. 


142  HOW  TO  TEACH  AEITHMETIC 

Morra.  This  is  a  very  old  Eoman  game  and  is  played 
with  great  enthusiasm  by  Italians,  old  and  young.  A 
group  of  children  stand  or  sit  in  a  circle.  Each  extends, 
at  a  given  word,  all  or  any  of  his  fingers.  An  immediate 
estimate  is  made  as  to  the  total  number.  All  are  added 
then  to  see  who  is  nearest  right. 

Roll  the  Hoop.  Draw  a  wheel  on  the  board,  or  use  an 
actual  wheel.  Place  number  combinations  for  spokes,  and, 
by  stating  results  in  order,  play  that  you  make  the  wheel 
1 ' turn"  as  fast  as  you  can. 

The  use  of  games  opens  the  way  for  children  to  handle 
the  objects.  Instead  of  being  passive  listeners  with  the 
teacher  manipulating  the  objects  as  she  demonstrates  a 
solution,  they  thus  become  active  participants  in  the 
process. 

Formal  and  Rational  Methods 

Objects  and  games  in  the  primary  grades  were  at  first 
but  the  remote  evidence  of  the  quickening  effect  of  critically 
scientific  methods,  which  had  filtered  down  from  the  higher 
institutions  through  the  grades  in  diluted  form.  Primary 
teachers  were  not  keen  in  grasping  the  new  idea ;  they  did 
not  see  that  a  wider  use  of  well  selected  materials  would 
assist  them  in  correcting  the  dogmatic  methods  of  in- 
struction then  in  vogue.  This  was  no  new  attitude  on  the 
part  of  the  teachers ;  as  a  class  they  have  never  solicited 
new  subjects  and  have  seldom  warmly  welcomed  new  meth- 
ods. These  things  have  come  in  under  the  influence  of  out- 
side pressures  that  could  not  be  withstood.  In  this  par- 
ticular instance  the  lower  schools  were  held  in  the  grip  of 
memoriter  methods,  facts  were  learned  with  but  the  mini- 

For  an  instructive  and  suggestive  list  of  number  games  and  rhymes 
see  Teachers'  College  Eecord,  November,  1912,  published  by  Columbia 
University. 


PRIMARY  ARITHMETIC  143 

mum  attempt  at  rationalization.  The  employment  of  induc- 
tion in  the  more  strictly  scientific  fields  spread  across  and 
produced  as  its  counterpart  " developmental' '  instruction 
in  the  grades.  As  the  currents  of  imitation  spread  from  the 
higher  to  the  lower  schools,  primary  teachers,  not  yet  con- 
scious of  the  modes  of  inductive  thinking,  seized  upon  its 
most  obvious  and  material  features,  and  began  to  use 
objects  for  purposes  of  illustration.  It  was  not  until  later 
that  teachers  sought  to  have  children  rediscover  or  invent 
through  inductive  thinking  or  developmental  instruction 
the  valuable  things  the  race  has  discovered  or  invented. 
Under  the  influence  of  this  idea  it  was  not  uncommon  for 
teachers  by  a  skillful  use  of  the  question  and  answer  device 
to  do  practically  all  the  thinking  for  the  class.  Another 
and  far  more  unfortunate  result  was  the  practice  of  mul- 
tiplying objective  illustrations  of  a  fact  long  after  the  chil- 
dren understood  the  fact.  The  attempt  to  rediscover  every 
relation  and  every  feature  about  a  number  fact  by  object 
teaching,  was  as  deadening  as  the  more  purely  memoriter 
methods  which  made  but  an  indifferent  attempt  at  relating 
facts  to  the  experiences  of  children. 

Emphasis  in  instruction  has  changed  from  the  mass  to 
the  group,  from  the  class  to  the  individual,  from  the  trans- 
mitting of  vicarious  experience  to  face  to  face  experience, 
from  a  single  appeal  to  a  variety  of  appeals.  The  edu- 
cational center  of  gravity  has  shifted  from  subject  matter 
to  child.  The  teacher  thinks  less  of  her  function  as  a  trans- 
mitter of  knowledge  by  repetitious  exercises  and  more  of 
the  necessity  of  breathing  personality  into  dead  materials, 
of  multiplying  the  points  of  contact  between  the  materials 
and  the  minds  of  the  children,  of  permitting  children  to 
learn  through  their  own  experiences  rather  than  through 
the  dogmatic  statement  of  book  or  teacher.  The  teacher 
tries  to  stimulate  a  feeling  of  need  in  the  minds  of  the 


144  HOW  TO  TEACH  AEITHMETIC 

children.  True  leadership  as  expressed  in  educative  meth- 
ods rather  than  coercive  methods  characterizes  the  spirit 
of  modern  instruction.  Drill  as  shown  in  another  chapter 
is  rational  rather  than  formal. 

Counting 

Attention  will  now  be  directed  to  certain  specific  phases 
of  arithmetic.  The  most  fundamentally  important  thing 
for  beginners  is  to  learn  to  count.  Counting  consists  in 
thinking  objects  serially  related  and,  later,  in  thinking  the 
symbols  that  represent  those  relationships.  Consequently, 
it  is  not  synonymous  with  measurement,  for  that  consists 
of  comparing  or  measuring  objects  alike  in  nature — as 
blocks,  two  lines,  two  sides  of  a  room,  some  shorter,  some 
longer.  A  quantitative  idea  of  the  world  of  nature  can  be 
gotten  only  by  comparing  and  measuring,  but  the  most 
elemental  ideas  of  number  are  gotten  by  counting.  The 
most  fundamental  mathematical  concept  depends  upon  a 
recognition  of  the  one  to  one  correspondence  between 
objects.  The  expression  of  such  correspondences  is  count- 
ing. Counting  is  as  old  as  the  race.  Primitive  people  use 
a  variety  of  objects  in  counting — pebbles,  sticks,  stones, 
toes,  hands,  fingers,  shells,  grains  of  corn,  notches  on  sticks, 
knots  in  strings.  Their  idea  of  number  was  expressed 
through  the  correspondence  of  the  thing  with  the  number 
of  notches  (cut  in  a  stick)  or  knots  (tied  in  a  string),  and 
names  to  express  the  correspondence  were  limited  or  want- 
ing. "With  the  invention  o~f  names  counting  began. 

The  facts  of  number  are  fixed  through  counting.  Chil- 
dren learn  that  17  is  less  than  18  and  more  than  16.  In 
other  words,  the  place  relations  of  numbers  are  fixed  by 
counting ;  particularly  if  the  pupils  learn  to  count  forward 
and  backward.  The  operations  of  addition,  subtraction, 


PRIMARY  ARITHMETIC  145 

multiplication,  and  division  are  attempted  refinements  of 
complex  forms  of  counting. 

Number  Symbols 

The  next  problem  after  children  have  acquired  the  num- 
ber concepts  through  the  use  of  objects  is  that  of  teaching 
them  to  read  and  write  number  symbols.  The  problem 
that  the  teacher  has  at  this  point  is  precisely  the  one  she 
has  in  teaching  beginning  reading.  The  pupils  already 
recognize  by  ear  the  number  names,  and  are  reasonably 
facile  in  communicating  them  by  speech,  but  they  do  not 
know  the  figures  by  sight  and  cannot  write  them.  Sight 
recognition  must  precede,  logically  and  chronologically, 
the  writing  of  the  symbols.  At  the  beginning  there  must 
be  a  definite  association  of  the  symbol  and  the  written 
form  with  the  idea  through  objects.  This  association  has 
usually  been  made  by  the  children  themselves,  so  that 
formal  instruction  of  this  character  may  not  be  needed  at 
all.  Those  courses  of  study  that  set  the  limits  from  0  to  10 
for  the  first  term  or  year,  and  from  10  to  20  the  second 
term,  and  20  to  100  the  third  term,  are  not  only  mechanical 
and  illogical  but  they  destroy  interest,  because  they  do  not 
take  advantage  of  the  mental  equipment  of  the  children. 
It  is  contrary  to  the  tenets  of  psychology  to  expect  one  to 
have  the  same  interest  in  a  skill  that  has  already  become 
automatic.  The  process  of  reading  and  writing  figures  is 
interesting  in  itself.  The  prevailing  tendency,  however, 
is  to  associate  the  number  series  with  a  game  or  some 
interesting  activity.  If  any  instruction  in  the  place  rela- 
tions of  numbers  apart  from  objective  associations  is  neces- 
sary, it  should  follow  the  objective  presentation  of  the 
numbers.  Instantaneous  recognition  and  response  should 
follow  the  display  and  perception  of  figures. 


146  HOW  TO  TEACH  AEITHMETIC 

Notation 

The  Arabic  notation  is  a  method  of  combining  numbers, 
giving  them  value  according  to  the  place  they  occupy, 
letting  each  figure  indicate  ten  times  as  much  as  the  one 
to  the  right.  It  is  a  simple  device,  an  artificial  method 
of  giving  to  a  few  figures  a  remarkable  power.  The 
reading  of  numbers  of  large  denominations  presupposes  a 
knowledge  of  notation.  When  pupils  have  learned  the 
number  language  up  to  ten,  this  may  be  extended  rapidly 
from  ten  and  one  to  ten  and  nine,  and  from  two  tens  to 
nine  tens  and  nine,  and  so  on.  When  they  can  count  and 
write  to  100,  the  teacher  may  safely  introduce  units,  tens, 
and  hundreds;  when  they  can  read  and  write  to  1,000,  she 
may  introduce  thousands,  and  so  on.  A  detailed  applica- 
tion is  seen  in  addition.  When  children  can  add  single 
columns  up  to  nine  or  less,  and  can  write  to  one  hundred, 
they  may  be  taught  the  adding  of  double  columns  without 
carrying.  In  teaching  them  the  reason  for  writing  one 
figure  under  another,  the  teacher  may  proceed  by  saying, 
"Let  us  see  how  many  tens  there  are  in  48."  The  class 
discovers  that  there  are  four  tens  and  eight  units;  this 
explains  why  48  is  written  as  it  is.  The  teacher  next  says, 
"I  want  to  write  41  under  48.  Let  us  see  how  many  tens 
there  are  in  it."  The  children  count  ten,  twenty,  thirty, 
forty — four  tens  and  one  unit.  It  is  then  easy  to  see  that 
if  we  want  to  add  all  of  the  ones  (units)  together  and  all 
the  tens  together,  the  most  convenient  arrangement  would 
be  to  write  them  in  separate  columns. 

Roman  Numerals 

Very  little  attention  should  be  given  the  Koman  numer- 
als, for  two  reasons:  (1)  They  are  rapidly  going  out  of 
use,  and  (2)  they  are  difficult  to  learn  and  more  cumber- 


PRIMARY  ARITHMETIC  147 

some  to  write  than  Arabic  numerals.  Some  years  ago 
Professor  Robert  M.  Yerkes  of  Harvard  conducted  a  num- 
ber of  experiments  to  determine  the  time  required  to  write 
Roman  numerals  and  the  number  of  errors  made  in  using 
them.  He  found  that  it  takes  three  and  one-third  times 
as  long  to  write  the  Roman  numerals  from  1  to  100  as  the 
Arabic,  and  the  chance  of  error  is  twenty-one  times  as 
great;  it  takes  three  times  as  long  to  read  the  Roman 
numerals  from  1  to  100  as  the  Arabic,  and  the  chances  of 
error  are  eight  times  as  great.  In  view  of  their  limited 
use,  there  is  no  excuse  for  teaching  any  Roman  numerals 
.except  those  from  one  to  twenty,  fifty,  one  hundred,  five 
hundred,  and  one  thousand. 


CHAPTER  XI 
THE  TEACHING  OF  THE  FUNDAMENTALS 

The  Five  Operations 

There  are  but  five  simple  operations  to  be  learned.  "Each 
of  them  involves  one  or  more  quantities  measured  by  some 
unit.  If  we  have  a  simple  quantity — for  example,  a  24-inch 
line — we  can  repeat  it  a  given  number  of  times,  say,  three ' 
times,  the  result  being  72  inches ;  or  we  can  divide  it  into 
a  number  of  equal  parts,  say,  three,  the  result  being  8 
inches.  If  we  have  two  quantities  measured  by  a  common 
unit,  we  can  combine  them,  thus  finding  how  many  of  the 
common  units  there  are  in  the  whole;  or  we  can  find  how 
many  more  of  that  unit  are  in  the  larger  than  in  the 
smaller ;  or  we  can  find  how  many  of  the  smaller  quantity 
there  are  in  the  larger.  All  of  arithmetic  consists  in  doing 
one  or  some  combination  of  these  operations,  and  it  is  essen- 
tial that  the  child  have  so  much  experience  in  handling 
objects  to  perform  them  that  he  shall  think  their  mean- 
ing and  know  better  than  to  try  to  multiply  5  inches  by 
3  inches,  or  to  add  5  and  4  inches. ' '  ^ 

Now,  the  society  for  which  our  schools  are  preparing 
does  not  demand  the  mastery  of  a  logically  arranged  series 
of  mathematical  principles.  It  does  demand  a  thorough 
knowledge  of  a  few  things,  and  their  applications.  For' 
example,  it  demands  ability  to  count,  to  read,  and  to  write 
numbers;  also  accuracy  and  rapidity  in  the  fundamental 
operations.  To  this  end  every  well-ordered  course  will 

1  Fiske  Allen,  ' '  Is  Arithmetic  a  Science  of  Numbers  or  of  Sym- 
bols? "  The  Kansas  School  Magazine,  Emporia,  November,  1912. 

148 


TEACHING  OF  THE  FUNDAMENTALS  149 

provide  an  abundance  of  systematic  drill  in  these  processes, 
a  drill  that  will  be  continued  throughout  the  upper  grades 
until  the  desired  standards  of  accuracy  and  rapidity  are 
secured. 

The  Serial  Relation  of  Numbers 

Some  of  the  best  courses  of  study  demand  entire  famil- 
iarity with  the  number  scale  to  120  before  any  work  in 
formal  addition  is  begun.  Care  must  be  taken  that  the 
transition  from  things  and  images  to  symbols  shall  be 
gradual.  The  counting  and  recognition  of  the  number 
scales  involves  knowing  that  any  number  and  1  more 
gives  the  next  number  in  the  scale,  and  that  the  number 
immediately  preceding  any  number  in  the  scale  is  one 
less  than  the  number,  and  that  one  less  than  any  number 
is  the  number  immediately  preceding  it.  The  school  soon 
affords  numerous  opportunities  for  teaching  counting; 
for  example,  the  number  of  children  in  the  room,  the 
group  or  class;  the  number  absent;  the  chairs  in  the 
room,  pencils  in  the  box;  the  selection  of  certain  objects 
from  other  objects ;  the  counting  of  objects  both  seen  and 
touched;  the  counting  of  sounds  (taps)  ;  the  repetition  of 
an  act.  This  may  be  followed  by  more  abstract  work  when 
children  count  forward  and  backward,  and  later  by  tens, 
fives,  twos,  and  fours.  It  is  clear  that  such  counting  in- 
volves both  addition  and  subtraction.  When  these  have 
been  begun  in  a  more  formal  way,  the  number  scale  may 
be  extended,  and  other  numbers  used  for  cumulative 
counting.1 

The  series  idea  which  is  the  basis  of  the  number  concept 
may  be  abstracted  very  early  in  the  school  experience  of 
the  average  child. 

It  is  unnecessary  to  teach  every  number  objectively.    A 

i  See  California  State  Course  of  Study. 


150  HOW  TO  TEACH  ARITHMETIC 

pupil  soon  comprehends  that  16  or  23  or  any  number  occu- 
pies a  certain  place  in  a  series.  He  does  not  think  of  it 
objectively.  For  this  reason  the  number  scale  should  be 
taught  and  memorized  as  such. 

The  structure  of  the  number  scale  may  be  shown  in  the 
grouping  involved  in  counting  by  tens. 

0  10  20  30  40,  etc.,  etc.    90          100 


1 

11 

21 

31 

41,  etc.,  etc. 

91 

2 

12 

22 

32 

42,  etc.,  etc. 

92 

3 

13 

23 

33 

43,  etc.,  etc. 

93 

4 

14 

24 

34 

44,  etc.,  etc. 

94 

5 

15 

25 

35 

45,  etc.,  etc. 

95 

6 

16 

26 

36 

46,  etc.,  etc. 

96 

7 

17 

27 

37 

47,  etc.,  etc. 

97 

8 

18 

28 

38 

48,  etc.,  etc. 

98 

9 

19 

29 

39 

49,  etc.,  etc. 

99 

Counting  by  ones  and  tens  may  be  extended  to  counting 
or  adding  by  twos,  three,  fours,  fives,  etc.  These  exercises 
lie  at  the  foundation  of  good  work  in  addition.  Oral 
drills  of  this  kind  should  be  given  until  all  sums  of  this 
character  can  be  given  accurately  and  rapidly. 

In  the  reading  of  whole  numbers,  do  not  allow  pupils  to 
use  "and,"  as  one  hundred  and  eighteen.  This  should  be 
read  one  hundred  eighteen.  The  "and"  will  be  needed 
later  to  indicate  the  decimal  point. 

Addition 

Although  addition  grows  out  of  counting,  care  must  be 
taken  to  avoid  counting  in  adding.  Both  the  oral  and 
written  forms  of  the  combinations  must  be  memorized  per- 
fectly. Drill  until  the  sums  can  be  given  at  sight. 

The  order  of  presentation  of  the  combinations  should  be : 


TEACHING  OF  THE  FUNDAMENTALS  151 

(1)  objects,  (2)  representation  of  the  combinations — ob- 
jects not  present,  (3)  recalling  the  combinations — values 
memorized,  (4)  concrete  problems  without  the  use  of 
objects.  Readiness  in  giving  the  result  of  any  two  num- 
bers should  be  considered  of  first  importance.  Attention 
must  be  called  to  the  fact  that  addition  is  the  process  of 
combining  units  of  the  same  denomination.  Such  Quan- 
tities as  dollars  and  gallons  cannot  be  added;  th  in 
only  be  set  down.  Such  quantities  as  yards,  feet,  ..id 
inches  can  be  changed  to  like  units. 

In  the  case  of  forty- five  of  the  eighty-one  addition  combi- 
nations which  require  automatic  mastery,  the  sum  is  not 
greater  than  ten.  These  forty-five  combinations  follow: 

123456789 
111111111 

2345678910 


1 

2 

3 

4 

5 

6 

7 

8 

2 

2 

2 

2 

2 

2 

2 

2 

3 

4 

5 

6 

7 

8 

-9 

To 

1 

2 

3 

4 

5 

6 

7 

3 

3 

3 

3 

3 

3 

3 

4 

5 

6 

f<;7 

8 

9 

To 

1 

2 

3 

4 

5 

6 

4 

4 

4 

4 

4 

4 

5 

6 

7 

8 

9 

To 

1 

2 

3 

4 

5 

5 

5 

5 

5 

5 

6 

7 

8 

9 

To 

152 


HOW  TO  TEACH  AEITHMETIG 


1      2 

6  6 

7  8 


3      4 

6       6 

9     10 


123 

777 

8       9     10 

1      2 

8       8 

9   To 
1 

9 

To 

A  convenient  way  to  express  addition  combinations  is : 
123456789 


2 

3 

4 

5 

6 

7 

8 

9 

10 

4 

5 

6 

7 

8 

9 

10 

11 

6 

7 

8 

9 

10 

11 

12 

8 

9 

10 

^11 

12 

13 

10 

11 

12 

13 

14 

12 

13 

14 

15 

14 

15 

16 

16 

17 

18 

TEACHING  OF  THE  FUNDAMENTALS  153 

There  are  other  ways  in  which  the  important  combina- 
tions may  be  related.  For  example,  there  are  the  old- 
fashioned  addition  tables: 

1  +  1  =  2 
1  +  2  =  3 
1  +  3  =  4 
1  +  4  =  5  etc. 

The  combinations  may  be  arranged  on  the  basis  of  the 
sums.  For  example : 

1+1  =  2  1  +  61 

1+2=3  2+5^=7 

i+»i.«  344/ 

2+2J-  1+71 

-  .  2  +  6  I  _a 

JJJU6  3  +  5  f 
4+4J 

1  +  51  etc. 

2  +  4  ^  =  6 

3  +  3J 


Column  Addition 

The  next  two  pages  are  taken  verbatim  from  the  course 
of  study  of  the  Los  Angeles  City  Schools : 

233 
654 

223          Use    the    combinations    now    mastered    in 
3       building  columns  for  '  adding.     An  examina- 
tion of  these  illustrative  columns  will  show 
2      £      how  columns  can  be  constructed  and  extended 
432      upward  as  far  as  the  teacher  likes. 
349 
2      53 

In  beginning  column  addition  with  children  in  the  pri- 


154  HOW  TO  TEACH  AEITHMETIC 

mary  grades,  place  the  following  column  on  the  board. 

Take  the  chalk,  and,  beginning  at  the  foot  of 

3  the  column,  say:    "Two,  three, — five,"  pointing 

to  the  numbers  as  named,  and  write  the  5  to  the 

g        32      right  of  the  3.    Then  say,  "Five,  four,— nine. ' ' 

23      Write  the  9  to  the  right  of  4.    Then  say,  "Nine, 

6        20     three, — twelve,"  and  write  the  12  to  the  right 

14     of  the  3.     Then  continue,  "Twelve,  two— four- 

12     teen,"  writing  the  14  to  the  right  of  the  2,  and 

K      so  on  until  the  column  is  added.    At  each  step 

have  the  children,  collectively  or  individually, 

repeat    after   you    each    statement.     Drill    the 

pupils  until  they  can  go  through  this  without  error.     If 

there  is  any  hesitancy  about  the  combinations,  point  to  the 

combination  above,  so  that  they  may  learn  where  to  find 

the  correct  form  if  they  should  forget. 

After  this  process  and  language  form  is  established, 
write  similar  columns  on  the  board  for  each  pupil,  with 
instructions  for  him  to  do  the  exercise  himself.  The 
teacher  should  pass  from  one  to  another,  hearing  each  give 
the  form.  As  a  pupil  finishes,  let  him  exchange  examples 
.with  another  pupil,  first  erasing  the  side  columns.  To 
avoid  confusion  it  is  well  to  write  two  or  three  examples 
in  excess  of  the  number  in  the  class,  so  that  no  pupil  need 
wait.  As  a  further  convenience,  it  may  be  helpful  for  the 
pupil  who  finishes  a  column  to  write  his  name  underneath 
it.  The  teacher,  passing  around,  later  erases  the  answer 
and  the  side  columns,  and  writes  "C"  (correct)  or  "X" 
(wrong)  after  his  name.  The  place  is  then  ready  for 
another  pupil. 

With  a  few  pupils  there  will  be  a  continual  tendency  to 
make  mistakes  in  the  left-hand  figure,  to  write  42  instead 
of  32,  etc.  This  means  that  insufficient  work  has  been  done 
on  the  number  scale.  Suppose,  as  in  the  illustration  given, 


TEACHING  OF  THE  FUNDAMENTALS  155 

the  pupil  writes  42  instead  of  32.    To  correct  this,  several 
methods  are  at  the  option  of  the  teacher. 

(1)   She  can  go  back  for  more  drill  in  the 
3  decades,  then  make  the  application  to  the  diffi- 

culty in  hand.     (2)  She  can  have  him  write,  in 
,~      ascending  column,  the  number  beginning  with 
23      23,  until  the  next  2  is  reached.     (3)   She  can 
6        20      draw  a  line  under  23,  and  ask,  "What  2  next 
14      above  23?"     (Answer,  "32.") 

After    the    combinations    already   mentioned 

5      have  been  mastered,  and  every  child  can  work 

out  the  side  columns  of  any  column  of  figures 

built  up  out  of  these  combinations,  readily  and 

without  mistake,  the  same  combinations,  in  their  reverse 

form,  should  be  treated  in  like  manner. 

The  purpose  of  this  is  to  drill  the  pupils  in  learning  new 
combinations  and  in  visualizing  the  end  figures  of  the  suc- 
cessive partial  sums.  After  this  form  has  been  mastered, 
the  teacher  should  continue  addition  without  writing  the 
sums  at  the  side,  and  train  the  pupil  to  add  without  this 
help.  In  starting  this  it  is  well  to  require  the  pupil  to  add 
directly,  thus:  five,  nine,  twelve,  fourteen,  twenty,  twenty- 
three,  etc.  If  he  makes  mistakes,  have  the  pupil,  in  imagi- 
nation, go  through  the  form  of  the  partial  sums  in  the 
side  column,  without  actually  writing  them.  First  at- 
tempts will  be  slow,  but  a  few  exercises  will  cause  him  to 
depend  upon  his  own  visual  imagining.  Proceed  in  the 
same  way  to  add  other  columns  in  review. 

In  all  this  early  work,  the  child  should  never  be  per- 
mitted to  perform  any  work  in  addition  at  his  seat,  but 
always  at  the  board,  in  full  view  of  the  teacher.  Children, 
if  allowed  ttie  time,  will  fall  back  into  the  habit  of  counting 
up  the  sums  serially.  It  is  a  mistake  to  think  that  children 
will  "outgrow"  this  habit,  once  it  ,is  formed.  Changing 


156  HOW  TO  TEACH  ARITHMETIC 

one's  habits  is  not  so  simple  a  matter  as  this.  To  prevent 
this  habit  from  being  formed,  the  teacher  must  first  give  in 
columns  only  those  combinations  which  the  children  have 
first  learned  thoroughly,  and,  second,  always  insist  that  the 
work  be  performed  at  the  board  and  in  full  view  of  the 
teacher.  Do  not  permit  the  child  to  stop  and  "think." 
He  either  knows  the  sum  or  not.  If  he  shows  the  least 
hesitancy  he  must  either  be  told  the  answer  or  be  permitted 
to  look  at  the  combination  involved  in  its  answer.  For 
this  purpose  the  combinations  should,  with  their  sums, 
always  be  written  on  the  board  in  full  view  of  the  child. 

Concert  Recitations  Versus  Individual  Tests 

Concert  work  is  good,  but  it  should  not  be  employed 
exclusively,  for  many  children  are  thereby  made  dependent 
in  their  work.  Again,  if  a  teacher  uses  it  too  generously, 
she  cannot  know*  what  the  individuals  are  capable  of  doing. 
In  addition  work,  the  teacher  must  keep  in  mind  the  fact 
that  her  class  will  not  proceed  uniformly  in  the  acquisi- 
tion of  the  work,  and  that  in  consequence  she  must  'provide 
some  way  to  give  much  individual  instruction.  The  prin- 
cipal of  each  building  should  keep  in  close  touch  with  each 
of  his  teachers  in  the  work.  To  do  this,  he  should  take 
individuals  from  the.  classes  into  the  office,  or  into  some 
convenient  room,  and  there  test  them  as  well  as  drill  them 
to  supplement  the  work  of  the  teacher.  He  should  know 
when  a  teacher  has  completed  the  study  of  a  given  group 
of  combinations,  and  determine,  through  taking  the  class, 
whether  it  is  ready  to  proceed  to  the  next  group. 

Each  successive  group  should  be  treated  in  the  same  way, 
except  for  the  side  columns,  which  may  be  discontinued. 
The  work,  however,  would  better  be  conducted  at  the  board, 
for  reasons  already  given.  The  teacher  should  prepare 
columns  of  figures  within  the  limit  of  the  particular  group 


TEACHING  OF  THE  FUNDAMENTALS  157 

which  she  is  treating,  then  dictate  these  to  the  class  at  the 
board.  Each  child  writes  the  column,  beginning  at  the 
bottom  and  going  toward  the  top,  and  of  course  adding  in 
the  same  direction,  in  order  that  the  combinations  may  be 
as  intended.  After  all  the  groups  have  been  studied  and 
columns  are  given,  with  the  combinations  arranged  hetero- 
geneously,  it  does  not  matter  whether  the  child  adds  up  or 
down.  In  fact,  it  is  a  good  check  on  the  work  to  have  him 
add  both  ways.  In  this  dictation  work,  after  the  children 
have  obtained  a  sum,  several  should  be  called  on  to  add 
aloud;  then  several  called  on  to  add  the  same  columns, 
but  with  2  tens,  3  tens,  5  tens,  6  tens,  etc.,  prefixed  to  the 
lowest  number.  This  gives  drill  in  the  upper  reaches  of 
the  number  scale  without  the  additional  work  of  rewriting 
the  numbers.  Columns  should  continually  be  given  which 
incorporate  and  use  the  combinations  of  groups  already 
learned.  /  The  pupil's  advance  into  new  ground  should  be 
very  slow,  in  order  that  he  may  master  the  old  very  thor- 
oughly. The  chief  merit  of  this,  as  of  any  other  system 
used  in  teaching  combinations,  rests  in  its  thoroughness. 
The  child  must  pass  by  easy  and  carefully  graded  steps 
from  the  simple  to  the  difficult.  At  every  step  of  the  way, 
the  teacher  must  keep  well  within  the  powers  of  the  child. 
Men  succeed  in  this,  not  so  much  by  reason  of  past  failures 
as  because  of  past  successes.  We  like  to  do  the  things  we 
can  do  well.  Just  so  with  the  child ;  he  gets  a  pleasurable 
emotional  reaction  from  doing  things  at  which  he  is  success- 
ful. This  is  the  chief  value,  as  well  as  pleasure,  of  review 
work — it  perfects  technique,  and  becomes  pleasurable  in 
proportion  to  the  child's  success  in  the  doing  of  it.  1 

Motive  for  Speed 

At  first  the  emphasis  must  fall  on  accuracy  and  neatnesg, 
To  attempt  to  secure  too  much  speed  at  first,  leads  to  inac- 


158  HOW  TO  TEACH  AEITHMETIG 

curacy.  But  gradually,  as  the  work  becomes  more  reflex, 
"speeding-up"  exercises  should  be  given.  Here  is  a  place 
for  the  right  kind  of  emulation,  such  as  is  found  in  con- 
tests among  classes,  or  among  individuals.  There  is  a 
tendency  among  the  advocates  of  "soft  pedagogy"  to  dis- 
parage rivalry  in  the  schoolroom.  History  shows  us,  how- 
ever, that  this  motive  has  been  a  powerful  factor  in  every 
line  of  social  and  individual  progress.  Because  rivalry  has 
a  selfish,  anti-social  side,  it  does  not  follow  that  it  lacks  a 
noble  and  helpful  one.  It  is  not  well  to  foster  emulation 
to  the  extent  done  by  the  Jesuits,  who  went  so  far  as  to 
pair  off  all  the  boys  of  a  school,  making  the  individuals 
of  each  pair  rivals  in  everything  pertaining  to  school  work. 
It  can  be  used  safely,  however,  in  pitting  class  against  class, 
or,  if  tactfully  done,  individual  against  his  fellows  of  the 
same  class.  Within  these  limits  emulation  will  prove  itself 
a  powerful  schoolroom  incentive. 

Habituate  the  Carrying  Processes 

Early  in  the  work  with  the  groups,  numbers  of  three  or 
four  figures  each  can  be  dictated  if  the  teacher  desires  it. 
However,  to  do  this  without  introducing  unfamiliar  com- 
binations, the  teacher  must  think  out  the  numbers,  taking 
into  consideration  the  figure  to  be  "carried."  Efficiency 
in  adding  demands  that  the  processes  become  reflex.  The 
adult  mind,  when  adding  columns  of  figures,  or  when  sub- 
tracting one  number  from  another,  is  absolutely  devoid 
of  even  the  feeling  of  the  concrete.  To  begin  in  the  first 
grade,  tying  splints  into  tens,  and  these  tens  into  hun- 
dreds, is  interesting,  perhaps,  but  it  gives  no  working 
ability,  and  we  question  its  value  in  giving  so-called  insight 
into  number.  As  a  matter  of  fact,  the  mechanical  process 
of  "bringing  down  the  2  and  carrying  the  1,"  to  the 
child,  is  just  as  much  an  objective  thing  as  would  be  a 


TEACHING  OF  THE  FUNDAMENTALS  159 

bundle  of  splints,  and,  besides,  it  happens  to  be  right  along 
the  line  of  the  child's  future  as  well  as  present  need.  One 
does  not  need  to  know  anything  about  the  mechanism  of  an 
adding  machine  to  operate  it  successfully,  nor  of  a  watch 
to  read  the  dial  plate. 

Multiple  Column  Addition 

Single  column  addition  is  naturally  and  logically  fol- 
lowed by  two,  three,  and  four  column  addition,  and  as  soon 
as  numbers  can  be  written  and  read  to  ten  thousand  and 
above,  addition  may  be  rapidly  extended  to  any  number  of 
columns.  Double  column  addition  should  be  carried  on  at 
first  without  carrying.  Short  columns  like 

20        22        26        22        12        24 
30        30        30        32        21        32 

33        13 

should  be  used  at  first.  These  may  be  followed  by  longer 
columns  and  by  columns  that  involve  "carrying."  A  good 
device  for  teaching  "carrying"  and  also  for  securing  .accu- 
rate work,  recommended  and  employed  by  the  Civil 
Service,  is  herewith  illustrated: 

83245 
6278 
5312 
246 
7582 
23 
24 
14 
21 

_8 

102663 


160  HOW  TO  TEACH  ARITHMETIC 

Subtraction 

Addition  and  subtraction  are  so  closely  related  that  they 
may  be  taught  simultaneously,  particularly  if  the  additive 
or  Austrian  method  in  subtraction  is  employed.  It  advo- 
cates "subtracting  by  adding "  rather  than  by  "taking 
from"  or  by  "borrowing."  Subtracting  by  adding  is  the 
method  used  by  the  expert  cashier,  by  all  money  changers, 
by  the  business  world.  A  number  of  successful  plans  for 
teaching  subtraction,  such  as  the  complementary  method, 
the  borrowing  and  repaying  plan,  simple  borrowing,  the 
left-to-right  plan,  and  the  "Austrian"  method,  are  de- 
scribed by  Smith.1  The  most  recent  of  these  is  the  "Aus- 
trian or  the  making  change"  method;  it  dates  from  the 
sixteenth  century.  It  consists  in  finding  what  number 
must  be  added  to  the  subtrahend  to  make  the  minuend. 
If  6  is  to  be  subtracted  from  13,  one  thinks  what  number 
must  be  added  to  6  to  make  13.  This  plan  has  two  decided 
advantages:  (1)  it  avoids  the  necessity  of  learning  sepa- 
rate tables  for  addition  and  subtraction,  and  (2)  it  in- 
creases speed  and  accuracy  in  addition,  as  subtraction 
forms  are  immediately  converted  into  addition  associations. 
"The  meaning,  the  applicability,  and  the  visual  form  of 
addition  and  subtraction  are  still  different.  Only  the 
process  of  remembering  and  using  the  fundamental  opera- 
tions is  the  same." 

Ordinarily  pupils  should  be  required  to  master  only  one 
of  these  methods.  A  pupil  who  has  already  become  habit- 
uated to  a  certain  method  of  subtracting  in  one  school 
should  not  be  required  to  learn  a  new  method  when  he 
changes  schools. 

The  same  id£a  is  easily  applied  to  subtractive  processes 

i  Teachers'  College  Eecord,  Columbia  University,  1909. 


TEACHING  OF  THE  FUNDAMENTALS  161 

involving  "borrowing."  In  solving  the  following  exercise, 
for  example^  the  child  would  say, 

8341 
6456 

1885' 

"six,  5, — eleven;  six,  8, — fourteen.  Five,  8, — thirteen; 
seven,  :/, — eight. " 

The  criticism  has  been  made  that  the  addition  idea  can 
not  be  used  in  the  subtraction  of  fractions  and  in  the  sub- 
traction of  dates,  but  this  criticism  is  not  based  on  fact. 

The  subtraction  combinations  are  the  same  as  the  addi- 
tion combinations;  the  lower  figure  in  each  instance  being 
equal  to  or  smaller  than  the  upper  one.  A  simple  arrange- 
ment of  them  is : 

123456789 
111111111 

23456789 
22222222 

3456789 
3333333 

456789 
4*  4   4   4   4   4 

56789 
55555 

6789 
6666 


162  HOW  TO  TEACH  ARITHMETIC 

789 
777 

8      9 
8      8 

9 
9 

Another  arrangement  may  be  based  upon  the  minuend; 
as, 

999999999 
987654321 

88888888 
87654321 

7777777 
7654321 

666666 
654321 

55555 
5-4   3   2   1 

4444 
4321 

333 
321 


TEACHING  OF  THE  FUNDAMENTALS  163 

2      2 
2      1 

1 

1 


Every  device  indicated  under  addition  may  be  and  should 
be  used  in  teaching  subtraction.  We  list  two  additional 
ones  for  drill  work.  Supply  figures  omitted  : 

Addition 
662345251 

ion897129116 

Using  subtrahend,  ask  how  many  makes  the  minuend  ;  as, 

9      8     10     13    21      5 
_4    _3    _7    _6    -7    -3 


Multiplication 

In  multiplication  in  arithmetic  we  have  a  number  to  be 
taken  as  an  addend  a  given  number  of  times.  Multiplica- 
tion involves  three  numbers,  the  multiplicand  (the  number 
to  be  repeated)  ;  the  multiplier  (the  number  showing  the 
number  of  repetitions)  ;  and  the  product,  showing  the 
result. 

The  multiplicand  and  the  product  must  be  of  the  same 
denomination.  This  is  not  violated  by  such  statements  as 
$3x2  =  $6,  because  this  should  be  read  $3  multiplied  by 
2  =  $6. 

The  multiplication  tables  developed  slowly.     It  was  not 


164:  HOW  TO  TEACH  ARITHMETIC 

i 

the  custom  for  them  to  be  learned  by  the  student  until  after 
the  seventeenth  century.  Human  ingenuity  devised  a  num- 
ber of  interesting  schemes  to  make  it  unnecessary  to  master 
the  tables.  The  "Sluggard's  Rule"  illustrates  how  ele- 
mentary products  were  obtained.  If  one  wished  to  multiply 
8  by  6,  the  following  form  was  used : 

8x2 
6    4 

The  2  and  4  are  the  respective  deficiencies  of  8  and  6 ;  i.  e., 
10  -  8  =  2   and   10  -  6  =  4.      The   multiplication  when   com- 
pleted would  appear — 
8x2        The  four  of  the  result  is  6-2  or  8-4,  and  the 

8  of  the  result  is  4x2 
6    4 

4  8  Thus  8x6=48.  Such  a  scheme  requires  no  multi- 
plication beyond  five. 

Millions  of  peasants  in  Russia  to-day  secure  the  correct 
result  in  multiplication  without  knowing  any  of  the  tables. 
Any  table  may  be  arranged  to  show  the  relationship  be- 
tween addition  and  multiplication;  for  example,  the  table 
of  twos : 


2 

2 

2 

2 
2 

2 

2 

2 
2 
2 

2 
2 

2 
2 
2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

TEACHING  OP  THE  FUNDAMENTALS 


165 


The  following  will  be  found  a  convenient  arrangement 
in  connection  with  the  tables  and  in  counting  by  twos, 
threes,  fours,  fives,  etc. 

It  will  be  noted  that  the  figures  of  the  second  column 
(Figure  I)  represent  the  counting  by  twos.  This  is  also 
true  of  the  second  row.  Those  in  the  third  row  (or  column) 
ascend  by  a  constant  addition  of  3 ;  those  in  the  fourth 
by  4,  etc.  The  diagonal  from  A  to  C  contains  the  figures 
1,  4,  9,  16,  etc. 


FIGURE  i 


1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

3 

6 

9 

12 

15 

18 

21 

24 

27 

30 

33 

36 

4 

8 

12 

16 

20 

24 

28 

32 

36 

40 

44 

48 

5 

10 

15 

20 

25 

30 

35 

40 

45 

50 

55 

60 

6 

12 

18 

24 

30 

36 

42 

48 

54 

60 

66 

72 

7 

14 

21 

28 

35 

42 

49 

56 

63 

70 

77 

84 

8 

16 

24 

32 

40 

48 

56 

64 

72 

80 

88 

96 

9 

18 

27 

36 

45 

54 

63 

72 

81 

90 

99 

108 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

110 

120 

11 

22 

33 

44 

55 

66 

77 

88 

99 

110 

121 

132 

12 

24 

36 

48 

60 

72 

84 

96 

108 

120 

132 

144 

B 


166 


HOW  TO  TEACH  ARITHMETIC 
FIGURE  II 


.  1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

£12 

The  following  number  game  is  based  upon  the  above 
arrangement. 

Have  several  pupils  at  the  board  (or  seats)  each  prepare 
a  large  square  containing  the  small  squares  as  indicated  in 
Figure  II. 

The  teacher  says  "8."  The  pupils  write  this  either 
under  the  4  x  2  or  the  2x4.  The  teacher  says  ' '  40. ' '  This 
may  be  placed  either  under  5x8,  8x5,  4  xlO,  or  the  10  x  4 ; 
the  teacher  continues  to  state  numbers  and  the  pupils  put 
them  in  the  proper  squares.  The  pupil  who  first  fills  all 
of  the  squares  in  a  row  or  a  column  or  a  diagonal  wins  the 
game.  The  speed  with  which  the  numbers  are  announced 
by  the  teacher  should  be  adjusted  to  the  class,  but  care 
should  be  taken  to  insure  that  the  numbers  are  announced 


TEACHING  OF  THE  FUNDAMENTALS  167 

rapidly  enough  to  keep  the  pupils  working  near  the  limit 
of  their  ability. 

Several  variations  of  this  game  will  suggest  themselves 
to  most  teachers. 

Division 

Division  is  the  analytic  process  of  finding  one  factor 
when  the  other  is  known.  A  distinction  is  not  infrequently 
made  between  division  by  partition  and  division  by  com- 
parison or  measurement.  In  partition  a  unit  is  to  be 
divided  into  equal  parts,  the  number  of  which  is  known 
and  the  size  of  each  is  required.  The  following  is  a  prob- 
lem in  partition.  Seventy-two  books  of  the  same  kind  cost 
288  dollars;  how  much  was  that  apiece? 

Numerical  solution :   -^   of  $288  =  $4. 

Oral  analysis:  Each  book  cost  one  seventy-second  of 
$288,  or  $4. 

In  division  by  measurement  we  have  a  whole  to  be 
divided  into  equal  parts,  the  size  of  which  is  known  and 
their  number  required.  A  problem  in  division  by  measure- 
ment is:  A  mother  divided  8  cakes  among  her  children, 
giving  each  two,  how  many  children  has  she?  Numerical 

4  =  the  number  of  children, 
solution:  2  cakes) 8  cakes 

Oral  solution :  She  has  as  many  children  as  8  cakes  are 
times  2  cakes,  or  4. 

The  numerical  expression  for  measurement  or  division 
may  be  developed  from  subtraction  in  the  same  way  that 
multiplication  was  developed  from  addition.  Give  a  prob- 
lem involving  the  division  of  a  number  (say  18)  into 
equal  parts,  the  size  of  each  part  being  (say  6).  Show 
that  this  same  thing  may  be  expressed  more  briefly. 


168  HOW  TO  TEACH  ARITHMETIC 

18 


Short  Division 

Before  beginning  the  more  formal  work  in  division  a 
child  should  be  able  to  tell  instantly  how  many  times  2  is 
contained  in  every  number  up  to  20,  and  should  state  the 
remainder,  if  any.    The  same  principle  holds  for  3  and  30, 
4  and  40,  etc.    The  most  elementary  facts  of  short  division 
are  taught  in  the  learning  of  the  tables.    The  problem  grows 
a  little  more  complicated  when  the  dividend  is  composed  of 
several  digits.    But  when  the  pupil  is  ready  for  this  type  of 
division,  he  should  be  familiar  with  the  system  of  nota- 
tion in  use.    With  this  as  a  basis,  the  explanation  of  the 
division  of  6234  by  2  is  relatively  simple.     The  quotient 
for  purposes  of  illustration  may  be  expressed  as  3000  + 100 
+  15  +  2.     This  form  of  expressing  the  quotient,  however, 
should  not  be  encouraged  or  indulged  in  for  any  con- 
siderable length  of  time.     On  the  other  hand  every  pupil 
should  be  able  to  express  any  quotient  in  such  terms  when- 
ever occasion  demands  it.     Whatever  constitutes  current 
practice  gives  us  the  clue  to  the  form  to  be  used  in  teach- 
ing short  division.     In  the  problem  2)6234  there  is  only 
one  difficulty,  i.  e.,  the  division  of  2  into  30  or  into  3  tens 
and  4  units.     This  difficulty  is  more  apparent  than  real, 
for  this  very  division  has  already  been  taught  in  connec- 
tion with  the  tables.     The  truth  is  there  is  no  real  diffi- 
culty in  teaching  the  facts  and  processes  of  short  division, 
if  the  tables  have  been  properly  learned.     The  difficulty, 
such  as  there  is,  is  in  notation. 


TEACHING  OF  THE  FUNDAMENTALS  169 

112  The  procedure  is  properly  related  to  knowledge 

21)2352        already  known  if  one  begins  with  11  into  121 
21  and  gradually  enlarges  the  dividend.     A  list 

~25          of  examples,  prepared  by  Mr.  0.   T.  Corson, 
21          editor  Ohio  Education,  illustrates  our  point : 
"42 
42 

12221-11=?  1111 

36663-11  =  ?  11)12221 

61105-11  =  ?  11 

24442-11  =  ?  12 

48884-11  =  ?  11 

73326-11  =  ?  12 

85547-11  =  ?  11 

97768-11  =  ?  11 

109989-11  =  ?  11 

The  above  series  may  be  followed  by : 


13322- 

-12  =  ? 

26664- 

-12  =  ? 

39996- 

-12  =  ? 

53328- 

-12  =  ? 

79992- 

-12  =  ? 

93324- 

-12  =  ? 

106656- 

-12  =  ? 

119988- 

-12  =  ? 

These  divisors  might  well  be  followed  by  13,  14,  15,  16, 
18,  19. 

Concrete  Problems 

In  the  beginning  all  concrete  problems  should  involve 
only  one  of  the  four  fundamental  processes.  The  same 
kind  of  problem  should  be  continued  until  the  form  is 


170  HOW  TO  TEACH  AEITHMETIC 

fixed.  The  solution  of  written  problems  may  be  aided  by 
(1)  a  pictorial  or  diagrammatic  representation  of  the 
problem,  (2)  an  oral  interpretation  or  estimate  of  the  con- 
ditions of  the  problem  before  the  solution  is  attempted,  (3) 
the  simplifying  of  the  situation  by  using  smaller  numbers. 
Encourage  the  children  to  make  original  problems  illus- 
trating the  application  of  the  fundamentals.  Problems 
may  be  based  upon  the  prices  of  common  groceries :  beans, 
sugar,  coffee,  rice,  potatoes,  flour,  apples,  bread;  upon 
articles  of  clothing,  as  calico,  muslin,  linen,  silk,  collars, 
shoes,  gloves,  slippers,  buttons;  upon  playthings;  upon 
farm  products;  upon  household  furnishings,  and  the  like. 
It  will  stimulate  interest  if  the  pupils  are  encouraged  to 
keep  their  original  problems  classified  in  a  permanent 
note-book. 


CHAPTEE  XII 
DENOMINATE  NUMBEES 

The  knowledge  of  numbers  was  probably  first  used  in 
measuring  objects.  It,  perhaps,  was  almost  as  difficult  to 
evolve  units  and  standards  of  measures  as  it  was  to  evolve 
a  system  of  numbers.  One  of  the  earliest  units  of  measure 
must  have  pertained  to  value.  Until  some  common  meas- 
ure of  value  was  agreed  upon  it  was  impossible  to  com- 
pare the  possessions  of  one  person  with  those  of  an- 
other, or  to  engage  in  trade  upon  any  equitable  basis. 
The  very  necessities  of  exchange  where  the  wealth  or  prop- 
erty of  some  men  consisted  of  furs,  of  others  of  cattle,  of 
still  others  of  tea,  ornaments,  tools  or  weapons,  required 
a  medium  so  that  the  value  of  one  could  be  expressed  in 
terms  of  the  other.  Occasionally  this  medium  or  standard 
was  purely  imaginary.  John  Stuart  Mill1 ,  informs  us  that 
certain  African  tribes  calculated  the  value  of  things  in  a 
sort  of  money  of  account,  called  macutes.  One  object  is 
worth  ten  macutes,  another  twenty,  and  so  on.  But 
mascutes  have  no  real  existence;  they  are  imaginary  con- 
ventional units  used  for  comparing  one  object  with  another. 

Money  the  Common  Medium  of  Exchange 

But  it  was  not  always  possible  to  exchange  one  object  of 
value  for  another  object  of  value.  Distance  must  have 
made  it  impossible  many  times  for  the  transfer  to  have 

been  made  on  the  spot.    A  medium  of  exchange,  a  currency, 
> 

l^i  Political  Economy,  chapter  5,  p.  11. 

171 


172  HOW  TO  TEACH  AEITHMETIC 

became  a  necessity.  Usually  gold  or  silver  served  as  money. 
The  reason  for  this  is  simple.  As  soon  as  people  were 
assured  of  the  necessities  of  life,  they  naturally  sought 
after  the  more  precious  and  the  more  ornamental  things, 
like  gold,  silver,  and  precious  stones.  These  are  valuable 
because  they  are  rare,  imperishable  and  portable.  Gold 
and  silver  became  the  basis  of  exchange  rather  th#n  jewels 
because  of  their  greater  durability. 

Money  became  both  a  unit  of  measure  and  an  equivalent 
of  value.  Because  it  was  a  measurer  and  because  objects 
varied  in  value,  divisions  and  subdivisions  of  the  unit  were 
necessary. 

It  must  be  remembered  that  money  was  first  weighed, 
not  counted.  Down  to  the  time  of  mediaeval  England 
coins  were  used  as  weights.  This  accounts  for  the  English 
pound  as  a  unit  of  weight.  The  Roman  pound  or  libra 
was  determined  by  a  balance  called  the  libra.  Thus  a 
libra  of  gold  could  be  converted  in  a  number  of  coins,  or 
so  many  coins  equalled  a  libra  of  gold.  The  standard 
libra  varied  with  different  Emperors,  but  in  mediaeval 
England  a  pound  Troy  came  to  mean  the  weight  of  240 
pennies  or  denarii.  The  history  of  the  names  employed 
in  the  various  coinage  systems  would  be  interesting,  but 
it  would  take  us  too  far  afield.  Any  adequate  account  of 
the  endeavors  to  prevent  the  corruption  of  coins,  the  de- 
preciation of  their  value,  abuses  due  to  dishonest  moneyers 
and  exchangers,  the  attempts  to  secure  a  nation-wide  or  a 
universal  system  of  coinage,  the  introduction  of  a  paper 
equivalent  for  gold  or  silver,  would  require  more  time 
and  space  than  we  can  afford  to  give. 

Troy  and  Avoirdupois  Weights 

The  Hindus  had  two  original  units  of  weight,  the  most 
ancient  of  which  was  the  Gerah,  whose  equivalent  was 


DENOMINATE  NUMBERS  173 

twelve  grains  of  barley.  The  seed  of  a  creeping  plant 
called  the  rati,  was  frequently  used  as  the  alternative  to 
the  barley-grain.  The  carat,  a  bean  of  an  Abyssinian 
tree,  which,  like  the  rati-seed  was  of  almost  uniform  size, 
was  also  used  as  a  measure  of  weight  by  dealers  in  precious 
metals. 

The  Roman  pound  consisted  of  5204  grains,  and  each 
pound  was  divided  into  12  unciae  or  ounces.  Henry  III,  in 
the  thirteenth  century,  decreed  that  "an  English  penny, 
called  a  sterling,  sound  and  without  clipping,  shall  weigh 
32  wheat  corris  in  the  midst  of  the  ear;  and  20  pennies 
do  make  an  ounce ;  and  12  ounces  one  pound. ' '  There  is 
no  need  of  additional  evidence  to  show  that  the  primary 
unit  of  weight  was  a  grain,  sometimes  barley,  sometimes 
rati-seed,  sometimes  wheat. 

Historic  Relation  of  Weight  and  Capacity 

The  measurements  of  weight  and  of  capacity  have  always 
been  closely  related.  Grains  and  vegetables  are  bought 
in  one  place  by  the  bushel  or  the  gallon,  and  in  another 
place  by  the  pound.  People  experience  little  difficulty  in 
translating  one  of  these  into  the  other.  The  expression  of 
this  relationship  between  these  two  types  of  measure  re- 
sulted in  a  statute  during  the  reign  of  Henry  VII,  which 
statute  reads: 

"One  sterling  (or  penny)  shall  be  the  weight  of  32 
corris  of  wheat  that  grew  in  the  midst  of  the  ear,  accord- 
ing to  the  old  laws  of  the  land."  The  table  evolved  from 
the  statute  was: 

32  grains 1  sterling 

20  sterlings 1  ounce 

12 ounces. . . .  „ 1  pound 

8  pounds 1  gallon 

8  gallons 1  bushel 


174  HOW  TO  TEACH  AKITHMETIC 

Here  we  find  the  two  types  of  measure,  weight  and 
capacity,  clearly  expressed  in  a  single  table. 

It  was  found,  however,  that  all  gallons  or  bushels  did 
not  weigh  alike.  They  varied  with  reference  to  the  kind 
of  material,  its  dryness,  and  whether  the  measures  were 
heaped  or  struck.  Hence  it  was  found  to  be  necessary 
to  state  more  accurately  the  meaning  of  a  pound. 
During  the  reign  of  Edward  II  the  term  avoir  du  pois  was 
invented  to  refer  to  a  uniform  pound  of  goods,  whether 
heavy  or  light.  Practice  more  than  legislation  resulted  in 
differentiating  Troy  and  Avoirdupois  weights;  the  former 
came  gradually  to  be  used  to  express  a  scale  of  relation  fcr 
precious  metals ;  the  latter  for  all  common  materials. 

Although  a  Troy  pound  is  supposed  to  consist  of  57(0 
grains  .and  an  Avoirdupois  pound  of  7000  grains,  so  that 
7000  grains  Troy  equal  1  pound  Avoirdupois,  it  is  quite 
generally  known  that  the  common  grain  is  no  longer  used 
as  the  base  unit.  About  the  middle  of  the  nineteenth  cen- 
tury the  British  Government  adopted  the  length  of  a  sec- 
ond's pendulum  oscillation  as  the  invariable  unit  of  length. 
Of  the  39.1393  parts  into  which  it  was  divided  the  British 
yard  contains  36.  Each  of  these  is  the  British  inch.  The 
distilled  water  which  fills  a  cubic  inch  is  found  to  w  eigh 
always  the  same  amount,  under  the  same  conditions.  This 
weight  is  divided  into  252.422  equal  parts;  the  British 
pound  avoirdupois  contains  7000  of  these  parts. 

"  Again,  it  is  possible  to  give  the  quantity  of  distilled 
water  which  weighs  exactly  10  of  these  pounds  avoirdupois. 
This  quantity  of  water  always  fills  the  same  space.  The 
space  is  the  ' content'  of  the  British  gallon,  the  unit  of 
capacity.  With  the  final  precaution  that  the  gallon  vneas- 
ure  shall  be  circular  at  base,  not  rectangular,  and  of  given 
diameter,  our  units  of  capacity  and  weight  are  clearly 
connected  and  rest  upon  the  firm Jbasis  of  the  'oscillation' 
unit  of  length." 


DENOMINATE  NUMBEKS  175 

Step  by  step  the  two  tables  were  differentiated.  The 
Troy  scale  being  expressed  as  follows : 

24  grains  make  1  pennyweight 

20  pennyweights  make  1  ounce 

12  ounces  make  1  pound 

Hence  5760  grains  make  1  pound  Troy 

While  7000  grains  make  1  pound  Avoirdupois 

And  the  Avoirdupois  scale : 

16  drams  make  1  ounce 
16  ounces  make  1  pound 
112  pounds  make  1  hundred-weight 
20  hundred-weights  make  1  ton 

The  dram  or  drachm  was  adopted  by  Koman  physicians 
instead  of  the  denarius.  The  most  curious  part  of  the 
Avoirdupois  was  the  use  of  112  pounds  to  equal  one  hun- 
dred-weight. Because  the  aiftount  of  material  necessary 
to  make  100  pounds  varied  with  localities  and  with  ma- 
terials, the  value  of  one  hundred-weight  varied.  Even- 
tually during  the  reign  of  Elizabeth  it  was  fixed  arbitrarily 
at  112  pounds. 

Length^  Surface  and  Solidity 

Units  of  length  are  derived  mainly  from  parts  of  the 
human  body  or  from  movements  described  by  them.  The 
nail,  palm,  foot,  and  handsbreadth,  clearly  refer  to  parts 
of  the  body.  An  ell  represents  the  distance  from  the  el- 
bow to  the  tip  of  the  longest  finger;  the  yard,  from  the 
armpit  to  the  tip  of  the  longest  finger ;  a  span,  the  distance 
between  the  extended  tips  of  the  thumb  and  the  middle 
finger;  the  pace,  two  steps;  the  mile,  a  thousand  paces. 

Units  of  length  and  of  weight  had  a  common  origin. 


176  HOW  TO  TEACH  ARITHMETIC 

"With  the  Hindus  3  barleycorns  placed  end  to  end  or  8 
placed  side  by  side,  measured  the  length  of  the  longest 
finger  joint,  which  distance  we  call  an  inch.  It  seems  quite 
probable  that  this  notion  prevailed  with  the  Arabians,  and 
that  the  Romans  related  12  such  parts  to  the  length  of  the 
foot.  At  any  rate,  by  the  time  of  Edward  II  in  England 
the  relationship  of  various  units  of  length  was  expressed 
in  this  table : 

3  barleycorns  (sound  and  dry)  placed  end  to 

end  make  1  inch 
12  inches  make  1  foot 

3  feet  make  1  yard 
5J  yards  make  1  perch 

There  is  a  tradition  to  the  effect  that  Edward  III 
had  the  length  of  his  arm  registered  in  a  bar  of  metal, 
and  this  was  divided  first  into  three  equal  parts,  and  each 
of  these  into  12  inches.  It  is  clear  that  different  people 
would  use  different  parts  of  the  table  differently.  Per- 
sons dealing  with  short  measures,  the  cloth  merchants,  for 
example,  would  become  proficient  with  the  yard,  foot  and 
inch;  masons  and  carpenters  would  frequently  require  a 
knowledge  of  both  the  short  and  middle  distance  units, 
while  agriculturalists,  soldiers  and  sailors  would  more 
likely  employ  the  longest  units. 

Square  measure  was  derived  directly  from  linear  meas- 
ure. It  really  does  not  require  a  separate  table,  as  the 
names  of  the  units  and  sub-multiple  in  each  instance  are 
the  same. 

Linear  Square 

12  in.  make  1  ft.  122  sq.  in.  make  1  sq.  ft. 

3  ft.  make  1  yd.  32  sq.  ft.  make  1  sq.  yd. 


DENOMINATE  NUMBEES  177 

The  similarity  of  terms  does  not  hold  beyond  this  point. 

Like  square  'measure,  cubic  measure  can  easily  be  de- 
duced from  linear  measure.  It  is  merely  an  expression  of 
length  in  three  directions. 

Uniformity  in  Measures 

"The  subject  of  weights  and  measures  of  a  people  bears 
much  the  same  relation  to  them  as  does  the  language  of 
ordinary  speech,  being  assumed  and  applied  in  their  daily 
occupations  without  active  thought  and  resisting  changes 
and  reforms  even  when  brought  about  by  the  most  strenu- 
ous efforts  and  with  convincing  proof  of  their  desirability 
or  necessity."  Wherever  civilization  has  progressed  and 
society  has  become  more  complex,  varying  and  inexact 
units  of  measure  have  been  found  unsuited  to  the  needs  of 
men.  Progress  in  civilization  always  necessitates  greater 
uniformity  of  the  units  of  measure  and  greater  exactness 
in  the  units  employed. 

When  the  grain  was  defined  to  be  "the  weight  of  a  grain 
of  wheat  taken  from  the  middle  of  the  ear  and  well  dried" 
it  was  a  variable  unit  and  all  those  units  of  weight  which 
were  defined  in  terms  of  the  grain  were  also  variable.  Our 
present  system  of  weights  and  measures  has  one  of  the  pre- 
requisites of  an  ideal  system — its  units  are  accurately  de- 
fined and  are  invariable,  but  it  is  a  poor  system  when 
judged  by  the  other  prerequisite — the  scale  for  ascending 
and  descending  reduction  must  be  uniform  in  a  given  table. 
The  Metric  system  has  both  of  these  prerequisites  and  is 
therefore  superior  to  the  English  system. 

The  necessity  of  adopting  standards  of  relations  in  the 
United  States  has  been  accompanied  with  a  variety  of 
problems.  The  different  states  have  had  no  uniform 
system  of  weights  and  measures.  To  obviate  this  diffi- 


178  HOW  TO  TEACH  AEITHMETIC 

culty  there  has  been  an  insistent  demand  for  the  reforma- 
tion of  the  weights  and  measures  and  for  the  establish- 
ment and  enforcement  of  a  national  system  of  standards. 

The  recognition  of  the  English  origin  and  variability 
of  colonial  measures  was  responsible  for  the  statement  in 
the  Articles  of  Confederation  that :  i  i  The  United  States  in 
Congress  assembled  shall  also  have  the  sole  and  exclusive 
right  and  power  of  regulating  the  value  of  alloy  and  coin 
struck  by  their  own  authority  or  by  that  of  the  respective 
states,  and  of  fixing  the  standards  of  weight  and  measures 
throughout  the  country. ' ' 

Later  the  Constitution  conferred  upon  Congress  the 
power  to  coin  money,  regulate  the  value  thereof  and  of 
foreign  coin,  and  fix  the  standard  of  weights  and  meas- 
ures.1 

After  careful  consideration  the  following  were  adopted : 
Th&  yard  of  36  inches,  the  avoirdupois  pound  of  7,000 
grains,  the  gallon  of  231  cubic  inches,  and  the  bushel 
of  2,150.42  cubic  inches.  The  standard  yard  was  the 
thirty-six  inches  comprised  between  the  twenty-seventh 
and  sixty-third  inches  of  a  brass  bar  prepared  for  the  court 
survey  in  London.  The  avoirdupois  pound  equals  1.215 
pounds  troy,  which  is  the  relative  equivalent  between  these 
weights  in  England.  The  units  of  measures  represent  the 
wine  gallon  of  231  cubic  inches  and  Winchester  bushel. 
These  units  were  adopted.  June  14,  1836,  and  the  secretary 
of  the  treasury  was  directed  to  supply  a  uniform  set  to  the 
governor  of  each  state.  Having  thus  provided  the  means 
of  uniformity,  Congress  delegated  the  responsibility  of 
enforcing  it  to  the  separate  states. 

The  metric  system  was  legalized  in  this  country  in  1866, 
and  is  now  in  daily  use  in  many  commercial  transactions. 
The  movement  to  substitute  it  for  our  national  system  is 
i  Article  1,  Section  8. 


DENOMINATE  NUMBERS  179 

not  powerful  enough  to  warrant  the  hope  of  its  speedy 
adoption. 

There  is,  however,  a  growing  sentiment  in  favor  of 
greater  simplicity  and  uniformity  as  to  the  method  in- 
volved in  selling  those  commodities  that  are  measured  by 
the  quantity  or  bulk.  This  sentiment  favors  the  selling 
of  such  products  and  articles  by  weight.  By  the  law  of 
1866  a  bushel  of  wheat  weighs  sixty  pounds,  of  corn  or 
rye  fifty-six  pounds,  of  barley  forty-eight  pounds,  of  oats 
thirty-two  pounds,  of  peas  sixty  pounds,  and  of  buck- 
wheat forty -two  pounds.  Duties  imposed  upon  these  prod- 
ucts are  based  upon  weights  rather  than  bulk  measure.  Of 
course  it  would  be  false  to  assume  that  each  of  these 
weights  equals  a  volume  equivalent  to  the  standard  bushel 
2150.42  cubic  inches.  These  weights  simply  represent  the 
averages  obtained  for  these  products  in  different  localities 
for  a  numbers  of  years. 

The  Teaching  of  Denominate  Numbers 

During  the  Middle  Ages  the  subject  of  denominate  num- 
bers was  of  great  importance  to  anyone  who  expected  to 
engage  in  commerce.  Most  of  the  cities  and  states  had 
their  own  systems  of  weights  and  measures  and  it  was  not 
uncommon  for  the  systems  of  two  neighboring  cities  to  be 
quite  unlike.  A  knowledge  of  the  most  important  sys- 
tems was  necessary  in  order  that  the  reductions  from  one  to 
another  could  be  readily  made.  The  arithmetics  of  the 
period  contained  numerous  tables,  and  the  lack  of  decimal 
fractions  necessitated  compound  numbers  of  several  de- 
nominations. 

Teachers  who  have  taught  some  years  are  familiar  with 
the  fact  that  much  of  the  material  formerly  included  in 
tables  of  measures  has  been  eliminated.  The  movement  for 


180  HOW  TO  TEACH  AEITHMETIC 

greater  simplicity  in  arithmetic  is  as  evident  in  denominate 
numbers  as  elsewhere.  The  modern  tendency  is  to  teach 
only  those  tables  that  are  generally  employed  by  the  ma- 
jority of  people  and  to  leave  the  technical  tables  of  tEe 
physician,  the  druggist,  and  the  jeweler  to  be  learned  by 
those  who  will  have  occasion  to  use  them.  The  newer  arith- 
metics have  eliminated  in  addition  to  Troy  and  Apothe- 
caries' and  Surveyor's  measures,  the  following: 

12  things  =  1  dozen  (1  doz.) 

12  dozen  =  1  gross  (1  gro.) 

12  gross  =  1  great  gross  (g.  gross) 

20  things  =  1  score 

24  sheets  of  paper -1  quire 

20  quires 

or        =1  ream 
480  sheets 

16  cubic  feet  =  l  cord  foot  (cd.  ft.) 
128  cubic  feet 
8  cord  feet 

6  "feet  =  l  fathom 
6086.7  feet  =  i  knot 
3  knots  =  1  league 


L=k  cord  (cd.) 


It  is  desirable  that  the  tables  should  be  collected  instead 
of  appearing  in  a  random  fashion  throughout  the  text. 

There  are  two  things  of  paramount  importance  to  be 
considered  in  the  teaching  of  denominate  numbers:  one  is 
the  reduction  or  changing  of  numbers  either  to  larger  or 
to  smaller  units,  and  the  other  is  the  automatic  mastery 
of  certain  simple  combinations  that  should  be  known  at 
sight.  In  teaching  the  reduction  of  denominate  numbers 
both  whole  numbers  and  fractional  measures  should  be 
used. 

Much  of  the  work  in  denominate  numbers  should  be  oral. 


DENOMINATE  NUMBERS  181 

Many  problems 'to  be  solved  at  sight  should  be  given.  In 
every  instance  they  should  be  related  to  the  daily  trans- 
actions of  business  life.  Long  written  problems  in  the  ad- 
dition, subtraction,  multiplication,  and  division  of  denomi- 
nate numbers  should  be  omitted.  Problems  involving  re- 
duction through  more  than  three  denominations  are  seldom 
used  in  the  business  world.  Children  should  be  made 
familiar  with  the  simple  and  necessary  facts  of  linear  meas- 
ure by  drawing,  measuring,  building,  counting,  and  group- 
ing ;  of  the  similar  facts  of  liquid  and  dry  measure  by  meas-, 
uring  liquids  and  a  few  common  products,  like  potatoes  or 
tomatoes,  and  by  applying  the  measures  to  products  bought 
for  home  use.  A  knowledge  of  the  other  units  of  measure 
should  be  acquired  and  applied  in  the  same  way.  As  far 
as  possible  children  should  have  the  actual  measures  pre- 
sented to  their  senses.  It  is  easy  to  make  the  mistake  of 
talking  about  these  things  without  children's  actually  hav- 
ing a  knowledge  of  them. 

Addition,  subtraction,  multiplication,  and  division  of 
denominate  numbers  are  based  upon  the  same  principles 
as  those  previously  developed  for  integers. 


CHAPTEE  XIII 

COMMON  FRACTIONS 

Common  fractions  have  been  regarded  as  one  of  the 
most  difficult  topics  in  arithmetic.  Many  young  teachers 
have  hesitatingly  approached  the  subject  and  have  grown 
discouraged  as  they  attempted  to  teach  children  who  were 
blindly  floundering  their  way  "through  fractions."  The 
difficulty  has  been  magnified  because  fractions  have  been 
taught  frequently  as  if  they  were  a  new  system  of  nota- 
tion, because  the  difficult  character  of  the  subject  has  sup- 
plied the  occasion  for  numerous  short  lectures  by  teachers 
to -pupils,  and  because  emphasis  has  been  placed  upon  the 
automatic  mastery  of  the  mechanical  phases  of  fractional 
relations,  while  the  rational  processes  have  been  ignored. 
To  have  it  constantly  dinned  into  your  ears  that  here  is  a 
new  and  very  hard  kind  of  work,  tends  to  increase  rather 
than  to  lighten  the  burden.  Small  wonder  that  teachers 
get  discouraged  and  pupils  perplexed ! 

The  last  of  the  reasons  assigned  above  has  been  re- 
sponsible for  a  most  unfortunate  result.  It  is  a  survival 
of  the  mediaeval  tendency  to  substitute  too  early  count- 
ing and  abstract  work  for  contact  with  objects.  When- 
ever this  is  done  the  number  sense  becomes  as  obtuse  as  it 
was  before  Pestalozzi  reintroduced  object  teaching.  The 
adult  mind  is  prone  to  break  fields  up  into  logical  classi- 
fications and  to  insist  upon  an  almost  verbal  mastery  of 
them,  neglecting  at  the  same  time  those  situations  that 
gave  birth  to  the  fact  and  in  which  the  fact  finds  its  pres- 
ent significance.  Children  do  not  see  the  world  broken  up 

182 


COMMON  FRACTIONS  183 

in  this  way.  They  grasp  general  truths  through  contact 
with  things.  They  must  have  objects  to  weigh,  to  measure, 
to  evaluate,  to  manipulate.  It  is  through  the  handling  of 
them  that  the  interpretative  powers  of  children  receive 
their  initial  training. 

Stages  in  Teaching  Fractions 

There  are  three  clearly  recognizable  and  well-defined 
stages  in  the  teaching  of  fractions.  Their  order  is  (1)  the 
meaning,  (2)  the  recognition,  and  (3)  the  manipulation 
of  fractions.  The  language  used  in  fractions  does  not  al- 
ways convey  the  intended  meaning  to  the  pupil.  The 
difficulty  is  not  due  so  much  to  language  as  to  the  vague 
idea  the  average  pupil  has  of  fractions.  The  language  of 
fractions  must  be  so  phrased  that  attention  is  fixed  upon 
the  notion  that  fractions  are  concrete  numbers  instead  of 
abstractions  ingeniously  devised  to  be  manipulated.  The 
limited  knowledge  of  fractions  children  have  when  they 
come  to  school  has  been  acquired  from  the  handling  of 
objects  common  to  their  play  and  the  home.  They  do  not 
think  |  or  ^  as  symbolically  represented,  but  as  referring 
to  particular  things.  Their  idea  is  expressed  by  the  sep- 
aration of  a  unit  or  of  groups  into  equal  parts. 

Instruction  must  begin  with  the  knowledge  children  have. 
In  fractions,  exercises  may  be  given  in  having  a  child 
divide  some  object,  say  an  apple,  into  two  or  three  equal 
parts.  The  child  may  then  count  the  parts  and  tell  how 
many  he  has.  When  this  operation  has  been  repeated 
enough  times,  attention  may  be  directed  to  the  units  in- 
volved. |  of  an  apple  is  a  fraction.  The  whole  apple  is 
the  original  unit,  and  the  one-third  of  an  apple  is  a  frac- 
tional unit.  The  two  indicates  the  number  of  pieces  of  the 
apple  taken.  The  3  indicates  the  kind  of  units  taken. 

Thus  there  are  at  least  three  steps  in  acquiring  a  frac- 


184  HOW  TO  TEACH  AKITHMETIC 

tional  concept:  (1)  a  unit  divided  into  equal  parts,  (2) 
a  comparison  of  one  or  more  of  these  parts  with  the  origi- 
nal unit,  (3)  a  group  of  these  parts  thought  together.  The 
symbols  used  to  express  fractional  ideas  need  not  be  taught 
until  later. 

The  concepts  for  one-half,  one-third,  and  one-fourth 
should  be  taught  at  the  time  children  are  receiving  their 
early  instruction  in  integers.  The  constructive  work  of 
the  primary  grades  is  admirably  adapted  for  this  purpose. 
Folding  exercises  develop  ideas  of  two,  three,  and  four 
equal  parts  of  a  unit.  They  are  excellent  devices  for 
developing  the  habit  of  self-criticism  and  of  training  in 
accuracy.  There  are  other  means,  incidental  in  charac- 
ter, for  inculcating  the  simpler  fractional  concepts.  Frac- 
tions appear  in  separating  classes  into  divisions,  sessions, 
into  periods,  in  the  forming  of  lines  of  march,  the  use  of 
rulers,  in  the  distribution  of  materials.  Opportunities  to 
use  these  will  not  be  overlooked  by  the  industrious  and 
thoughtful  teacher. 

A  fact  not  always  made  clear  to  beginners  is  that  a  frac- 
tion is  as  truly  a  unit  as  an  integer.  There  are  both 
integral  units  and  fractional  units.  The  fraction  one-half 
may  be  separated  into  fourths,  and  the  fourth  into  six- 
teenths. Any  part  may  be  considered  as  a  unit  and  may 
be  divided  into  equal  parts  in  the  same  way  that  the 
integer  was  separated  to  get  the  part.  This  idea  is  easily 
developed  by  the  use  of  objects,  and  pictures. 

Definition  of  a  Fraction 

A  commonly  accepted  but  faulty  definition  of  a  fraction 
is  that  it  is  one  of  the  equal  parts  of  a  unit.  That  this 
definition  is  incomplete  and  unsatisfactory  is  seen  when  we 
apply  it  to  f  or  f.  These  are  more  than  "one"  of  the 


COMMON  FRACTIONS 


185 


equal  parts  of  a  unit.  No  definition  is  adequate  that  elimi- 
nates the  majority  of  the  cases  that  should  come  under  it. 
A  fraction  may  be  correctly  defined  as  one  or  more  of  the 
equal  parts  of  a  unit.  This  definition  covers  all  the  cases; 
it  includes  the  separation  of  unity  into  equal  parts,  or  of 
groups  of  integers,  for  example,  10,  which  may  be  divided 
into  still  smaller  but  equal  groups. 

Devices  in  Teaching  Fractions 

Although  we  have  advocated  the  teaching  of  fractions 
by  means  of  objects,  we  are  not  unmindful  of  the  fact  that 
the  amount  of  object  teaching  possible  and  desirable  varies 
with  the  age  and  intellectual  maturity  of  the  class.  No 
hard  and  fast  rule  should  be  followed,  for  classes  differ 
widely  as  to  their  ability  and  attainment.  Object  work 
should  be  followed  by  pictures,  lines,  circles,  squares,  and 
rectangles.  Fractional  notions  may  be  represented  by  the 
line  as  follows: 


1 

2 

2 

I 

1 
4 

i 

i 

4 

i 

1 

1 

8 

1 

8, 

1 

i 

1        1 

changing  fractional  forms  may  be  shown  by  the  circle. 
How  many  fourths  in  one-half? 
How  many  eighths  in  one-fourth  ? 
How  many  eighths  in  one-half. 

Needed  Elimiimiions 

It  is  unnecessary  to  extend  illustrations  of  the  possible 
uses  of  pictures  and  drawings.  In  one  of  the  newer  pri- 
mary arithmetics  more  than  two  hundred  pictures  of  lines, 


186  HOW  TO  TEACH  ARITHMETIC 

circles  and  squares  are  used  in  various  ways  to  teach  frac- 
tions. These  are  not  used  as  extensively  as  they  should  be. 
Teachers  should  confine  themselves  largely  to  those  frac- 
tions that  are  used  in  every  day  business  life.  Long  frac- 
tions, as  iVVViWWV  an<^  exceedingly  complex  fractions,  as 


!_     4     21  ^  2 

3  +  7  +  32  4£0+  15    ' 

2  +  23      3  or  1  1 

5     56  +  5  15  10 

and  other  unbusiness-like  forms  should  be  omitted.  In 
earlier  days  fractions  of  this  sort  were  believed  to  be  of 
peculiar  value  in  sharpening  the  wits  of  people.  This 
afforded  sanction  enough  for  their  use  in  the  past,  but  such 
a  reason  is  discredited  to-day. 

The  History  of  Fractions 

The  history  of  the  evolution  of  fractions  shows  that  modi- 
fications come  about  slowly.  The  oldest  known  mathemat- 
ical records  indicate  that  fractions  were  extensively  used 
at  an  early  date.  The  systems,  however,  were  cumbersome 
and  the  methods  peculiar  and  complicated.  This  partly 
accounts  for  the  tradition  that  fractions  are  difficult.  The 
work  of  Ahmes,  which  dates  back  to  2000  or  3000  B.  C., 
devotes  considerable  space  to  fractions.  Most  of  his  frac- 
tions had  unity  for  a  numerator.  He  expressed  his  frac- 
tions by  writing  the  denominator  and  placing  over  it  either 
a  dot  or  a  symbol  called  TO.  Whenever  a  fractional  value 
occurred  that  could  not  be  thus  expressed,  it  was  written 
as  the  sum  of  two  or  more  unit  fractions.  -|  was  regarded 
as  TV  +  -A-  ,  -9\  as  -fa  +  vfg-  +  Tfg-.  In  the  case  of  f  an 
exception  was  made  and  a  symbol  was  used  to  represent  it. 
The  Babylonians  used  sexagesimal  fractions  ;  that  is,  frac- 


COMMON  FEACTIONS  187 

tions  having  a  constant  denominator  of  60.  They  wrote  "the 
numerator  only,  placing  it  a  little  to  the  right  of  the 
ordinary  position.  The  Greeks  at  first  used  unit  fractions 
and  indicated  them  by  simply  writing  the  denominator 
with  a  double  accent.  Later  they  used  ordinary  fractions, 
writing  the  numerator  once  with  a  single  accent  and  the 
denominator  twice  with  a  double  accent.  They  would  have 
written  T9^-  as  9'  15"  15".  The  Eomans  used  duodecimal 
fractions ;  that  is,  fractions  having  12  for  the  denominator. 
The  inch  and  the  ancient  ounce  were  outgrowths  of  this. 
The  Hindus  used  the  general  fractions  and  indicated  them 
by  writing  the  numerator  above  the  denominator  without 
any  line  separating  them.  Thus  f  was  written  |.  The 
Arabs  introduced  the  separating  line.  In  the  oldest  Hindu 
arithmetic,  the  Liliwati,  the  multiplication  of  fractions  is 
indicated  by  writing  the  fractions  consecutively  without 
any  symbol  between  them.  Addition  of  a  fraction  to  an 
integer  was  shown  by  writing  the  fraction  beneath  the 
integer,  and  subtraction  was  shown  in  the  same  way,  with 

4 
a  dot  prefixed  to  the  fraction.     Thus  4  +  f  was  written  f; 

4 

and  4-f,  as  f.  The  problem,  "Tell  me,  dear  woman, 
quickly,  how  much  a  fifth,  a  quarter,  a  half,  and  a  sixth 
make  when  added  together,'7  appears  solved: 

1       1       1       1       67 

5      4      2      6      60 

That  the  subject  of  common  fractions  was  a  difficult  one 
for  the  race  is  indicated  by  the  fact  that  unit  fractions 
were  used  for  thousands  of  years.  During  the  Middle  Ages 
the 'term  "common"  was  used  to  distinguish  these  frac- 
tions from  sexagesimal  fractions.  Common  fractions  are 
called  "  vulgar "  fractions  in  England. 


188  HOW  TO  TEACH  AEITHMETIC 

Since  the  Arab  invasion  of  Europe  there  has  been  almost 
no  change  in  the  formal  representation  of  fractions.  Many 
of  the  rules  found  in  Liliwati  of  the  Hindus  remain  prac- 
tically unchanged  to-day.  But  fractions  have  increased 
vastly  in  importance;  they  are  a  common  possession  of  the 
common  people;  they  are  constantly  needed  in  our  every- 
day transactions.  Text-books  give  far  more  space  to  them 
than  formerly,  the  problems  are  more  real  and  the  instruc- 
tion more  skillful. 

Interpreting  Fractions 

One  of  the  chief  difficulties,  however,  yet  remains:  the 
difficulty  experienced  by  children  in  interpreting  clearly 
and  quickly  the  perplexing  forms  and  terms  used.  It  is 
to  this  end  that  we  so  insistently  urge  that  children  learn 
to  objectify  common  fractional  quantities,  halves,  fourths, 
sixths,  etc.,  only  after  they  have  first  learned  them  in 
connection  with  actual  objects.  When  the  pupils  have 
acquired  skill  in  recognizing  these,  they  should  be  required 
to  re-apply  them  to  objects  and  to  pictorial  representations 
of  objects.  Drawings,  because  they  are  subject  to  a  wider 
application  than  objects,  constitute  the  natural  transitional 
material  between  objects  and  abstract  work.  A  knowledge 
of  the  simpler  fractional  parts  may  be  facilitated  by  ar- 
ranging them  into  related  groups : 

1.  The  whole,  halves,  fourths. 

2.  The  whole,  halves,  foi^rths,  eighths. 

3.  The  whole,  thirds,  sixths,  twelfths. 

4.  The  whole,  thirds,  fourths,  and  twelfths. 
(Suggested  by  the  Indianapolis  Course  of  Study.) 

1 1  In  each  of  the  figures  show  the  whole ;  a  half ;  a  fourth. 
How  many  halves  in  the  whole?  Show  them.  How  many 
fourths  in  the  whole?  Show  them.  How  many  fourths  in 


COMMON  FRACTIONS 


189 


the  half?  Show  them.  Compare  in  as  many  ways  as  pos- 
sible, pointing  out  each  fractional  part  named:  the  whole 
and  ^ ;  the  whole  and  £ ;  \  and  J. 


Fig.  2 


Fig.  3 


Fig.  5 


Fig. 


The  Fundamentals  of  Fractions 

After  the  pupils  thoroughly  understand  and  appreciate 
what  a  fraction  is  and  have  learned  to  write  fractional 
quantities,  they  should  be  taught  to  reduce,  to  add,  sub- 
tract, multiply  and  divide  them.  It  is  best  to  begin  these 
operations  with  fractions  that  appear  as  products  in  the 
multiplication  table.  Next  in  importance  to  the  funda- 
mental concept  that  a  fraction  is  one  or  more  of  the  equal 
parts  of  a  unit,  is  the  idea  that  fractions  may  be 
changed  to  equivalent  fractions  by  multiplying  both  terms 
by  the  same  number.  It  requires  considerable  skill  to 
make  clear  the  notion  that  there  may  be  a  change  of  form 
without  a  change  of  value.  If  we  take  the  fraction  f  and 
multiply  its  terms  by  2,  we  get  f .  Now,  the  |  of  a  foot  is 
an  inch  and  a  half,  and  therefore  f  of  a  foot  is  6  inches  and 
6  half  inches,  or  9  inches ;  but  f  of  a  foot  is  9  inches,  there- 
fore .f  of  a  foot  equals  f  of  a  foot.  The  teacher  must 
show  that  this  principle  is  true  whatever  the  unit  of  meas- 


190  HOW  TO  TEACH  ARITHMETIC 

ure  may  be,  or  whatever  the  fraction  of  that  unit.  If  this 
is  well  done  the  pupils  have  learned  the  "secret"  of  re- 
ducing fractions  to  lower  or  raising  them  to  higher  terms. 
They  then  understand  that  the  same  fractional  value  may 
have  many  different  names.  The  table  listed  below  shows 
this  fact,  and  it  may  be  used  also  to  find  a  common 
denominator  for  different  fractions. 


O3 

•g 

ojojojojojojajGoaj 

tir|          r-j         ,-g-j         _r-j        r-j          c-j         ^         ,_g-j          r-j 

,S     5 

r§        ^ 

COQOC^OC<J^IOCOOO 

O        iH 

cq    -5 

1           2 
2           4 

34                   567                   89 
16        8^      ••TO'   T2~   T"4     •  •     T6    T"8 

10 
"20^     •  • 

11      12 
2~2     2^4 

I 
3 

23456 
6''         9        ••12'*15'-18 

7 
•  •      21" 

8 
•  •      24 

1          J, 

4          4 

234 
8        ••       ••      12      ••       ••      16      •• 

A  •• 

•  •  irir 

1 

2                              3 

•-.     fT                                                      "n  —  g- 

5         ••  • 

••      10                ••      15 

*T7     • 

1 

1                                      _2_                                    _3_ 

4 

6         •  • 

1 

.       3 

i 

1                                                               2 

21 

3 

8         •  • 

1 

8        16 

JL                                                            2 

•      24 

-T77       , 

•    •        ••          g          ••        •   .        ..        .    .        ..        ^g 

_2_ 

10 

2 
~o~a"  -    •   • 

1 
1   Q        •    . 

1 

2  2 

2 

The  preceding  table  was  not  given  to  be  memorized. 
There  are,  however,  some  common  fractions  that  occur  so 
frequently  in  business  that  every  one  should  know  them. 
If  the  preceding  equalities  have  been  clearly  taught  and 
apprehended,  pupils  will  appreciate  the  desirability  of 


COMMON  FRACTIONS 


191 


learning    equivalent    decimal    forms 
cents. 

Common  Decimal  Parts 

Fractions  Fractions  of  $1 

\                  .5  $.50 

i                  .25  $.25 

i                  .33*  $.33J 

\                  .2  $.20 


and    equivalent   per 


Per  cents 
50% 

25% 


§ 
i 


.12* 
.1 

.75 

.66| 

.4 

.6 

.8 

•83* 

.37* 

.62* 

.87* 


$.12* 

$.10 

$.75 

$.66| 

$.40 

$.60 

$.80 

$.83J 

$.37* 

$.62* 

$.87* 


20% 

16|% 

12*% 

10% 

75% 

66|% 

40% 

60% 

80% 

83*% 

37*% 

62*% 


Reduction  of  Fractions 

The  "  reduction  of  fractions  to  equivalent  fractions  hav- 
ing a  common  denominator "  is  necessary  before  unlike 
fractions  can  be  added  or  subtracted.  We  can  add  $,  f ,  f , 
as  they  are  all  of  the  same  denomination — sevenths.  We 
can  easily  see  that  to  add  f  and  f ,  we  have  only  to  change 
§  to  sixths.  Similarly  many  fractions  can  be  changed  to 
equivalent  fractions  by  inspection.  To  find  the  value  of 
f  +  f +  |  is  slightly  more  difficult,  but  not  essentially  dif- 
ferent in  principle.  In  addition  of  dissimilar  fractions  the 
student  is  confronted  with  the  problem  of  discoverng  some 
common  number  or  multiple  to  which  all  the  different  frac- 


192  HOW  TO  TEACH  ARITHMETIC 

tions  may  be  related.  A  simple  way  of  finding  this  com- 
mon multiple  is  by  finding  the  product  of  the  prime  fac- 
tors of  the  denominators,  using  each  factor  the  greatest 
number  of  times  it  appears  in  any  number.  In  the  example 
above  we  get  24.  Thus,  if  both  the  terms  of  f  be  multiplied 
by  6  we  get  ^f .  Similarly  the  other  fractions  may  be 
changed  to  equivalent  fractions  having  the  common  denomi- 
nator 24,  and  the  addition  performed. 

What  has  been  said  about  addition  applies  with  equal 
force  to  the  subtraction  of  fractions.  The  order  of  diffi- 
culty is  the  same;  first  teach  the  subtraction  of  similar 
fractions ;  then  of  dissimilar  fractions,  and  finally  of  mixed 
numbers.  Addition  and  subtraction  may  be  taught  sim- 
ultaneously. Pupils  in  the  sixth  grade  should  be  able  to 
state  instantly  the  result  of  such  examples  as :  4  +  ^ ;  i  +  i ; 
£-4  and  £-•£. 

Multiplication  and  Division 

In  teaching  multiplication  and  its  reverse  process,  divi- 
sion, the  procedure  should  be  (1)  multiplying  or  dividing 
a  fraction  by  a  whole  number,  (2)  a  whole  number  by  a 
fraction,  (3)  a  fraction  by  a  fraction,  (4)  a  mixed  number 
by  a  whole  number,  (5)  a  mixed  number  by  a  mixed 
uumber. 

The  principles  underlying  these  operations  are  few  in 
number  and  simple  in  nature.  They  usually  appear  as 
rules.  But  they  should  be  so  presented  that  self-discovery 
on  the  part  of  the  pupils  is  inevitable.  To  some  of  these 
principles  we  have  already  alluded;  for  example,  the 
principle  that  fractions  may  be  changed  in  form  without 
altering  their  value  by  multiplying  or  dividing  both  terms 
by  the  same  number.  A  second  equally  important  truth 
is  that  only  like  fractions  can  be  added  or  subtracted.  The 


COMMON  FRACTIONS  193 

third  and  fourth  principles  refer  to  the  ways  for  increas- 
ing the  value  of  a  fraction.  Multiplying  the  numerator  by 
an  integer  multiplies  the  value  of  the  fraction  by  making 
more  parts  in  the  fraction,  and  dividing  the  denominator 
by  an  integer  multiplies  the  value  of  the  fraction  by  mak- 
ing the  size  of  each  part  larger.  The  fifth  and  sixth  prin- 
ciples refer  to  the  ways  for  diminishing  a  fraction.  Divid- 
ing the  numerator  by  an  integer  divides  the  value  of  the 
fraction  by  decreasing  the  number  of  parts,  and  multiply- 
ing the  denominator  by  an  integer  divides  the  values  of  the 
fraction  by  making  the  parts  smaller. 

It  will  be  noted  that  the  expressions  multiplication  and  * 
division,  as  applied  to  fractions,  are  extensions  of  the 
ordinary  meanings  of  those  terms,  for  the  former,  in  its 
original  meaning,  implies  increase,  and  the  latter  decrease ; 
but  when  two  proper  fractions  are  multiplied  together  the 
product  is  less  than  either  of  the  factors,  and  when  one 
proper  fraction  is  divided  by  another,  the  quotient  is 
greater  than  either  the  dividend  or  the  divisor. 

There  are  several  explanations  for  the  inversion  of  the 
divisor  in  the  division  of  fractions. 

1.  Inverting  the  divisor  gives  the  same  result  that  is 
obtained  by  reducing  both  fractions  to  their  least  common 
denominator  and  dividing  the  numerator  of  the  dividend 
by  the  numerator  of  the  divisor. 

f.    3  _    8     •    1  5  _    8_ 
-f  -TO"-  2-0  ~lV 

The  same  idea  is  involved  here  as  in  the  division  of  $8 
by  $15 ;  the  result  is  rV- 
But  f xf  =  A;  therefore  f-f  =  fxf  =  T\. 

2.  If  the  divisor  were  3,  then  the  quotient  would  be  ^  of 
f ,  but  the  divisor  is  £  of  3 ;  hence  the  quotient  is  4  x  ^  of  f , 
which  is  |xf,  or  T\. 

3.  f-f | -Quotient. 


194  HOW  TO  TEACH  AEITHMETIG 

Therefore   f  Quotient  =  f. 

i  Quotient  =  £  of  £  =  &. 
f  Quotient  =  4  x  T^  =  -fa. 

4.  Reasoning  from  known  to  unknown. 


f  contains  f  ,  1  time. 
i  contains  J,  -J  time. 
1  contains  f  ,  f  times. 

f  contains  f  ,  f  x  f  times  ; 
therefore    f-7-f  =  iV 

This  is  one  of  the  most  difficult  for  children  to  under- 
stand. 

The  mastery  of  common  fractions  is  no  longer  attempted 
in  any  one  grade.  Some  attention  is  given  to  them  in  all 
of  the  grades,  but  most  of  the  formal  treatment  of  the 
subject  is  found  in  the  fifth  grade.  Complex  fractions  are 
omitted  in  most  treatments  of  the  subject.  It  is  wise  to 
teach  'the  operations  with  only  as  much  theory  as  seems 
necessary  to  a  clear  understanding  of  the  subject,  and 
pupils  should  not  be  required  to  memorize  and  repeat  the 
explanations.  It  is  necessary  that  teachers  understand 
thoroughly  the  reasoning  underlying  the  various  processes, 
but  it  is  not  desirable  that  we  attempt  to  force  this  mature 
reasoning  upon  the  pupils.  Results  should  be  gauged  by 
the  knowledge  and  skill  attained,  not  by  the  number  of 
pages  covered. 


CHAPTER  XIV 

DECIMAL   FRACTIONS 

Origin  of  Decimal  Fractions 

"The  invention  of  decimal  fractions,  like  the  invention 
of  the  Arabic  scale,  was  one  of  the  happy  strokes  of 
genius."1  The  decimal  fraction  is  a  comparatively  recent 
product  of  man's  ingenuity.  As  Napier  says:  "It  is  a 
surprising  fact  that  decimals,  so  simple  and  convenient, 
were  not  invented  until  after  so  much  had  been  attempted 
in  physical  research  and  numbers  had  been  so  deeply  pon- 
dered." Decimals  were  unknown  to  the  Greeks  and  the 
Romans,  whose  number  systems  did  not  utilize  place  value 
in  the  sense  in  which  we  use  the  term. 

It  is  not  known  in  what  way  decimal  fractions  actually 
originated.  There  are  two  theories  which  account  for  their 
origin.  The  advocates  of  the  first  theory  believe  that  they 
were  devised  as  a  shorthand  method  for  expressing  certain 
common  fractions.  They  assert  that  since  the  base  of  our 
number  system  is  ten,  someone  devised  or  invented  decimal 
fractions  as  a  transition  from  common  fractions.  If  this 
theory  is  correct,  and  if  we  follow  the  historical  develop- 
ment of  the  subject  in  our  teaching,  the  intimate  relation- 
ship to  common  fractions  should  be  emphasized.  This  is 
the  method  followed  by  some  text-books. 

Those  who  advocate  the  seccmd  theory  maintain  that 
the  subject  arose  directly  from  the  decimal  scale  for  in- 

i  Brooks,  "Philosophy  of  Arithmetic/7  p.  443. 

195 


196  HOW  TO  TEACH  AEITHMETIC 

tegral  numbers.  Since  numbers  diminish  in  value  from 
left  to  right  in  a  ten-fold  ratio,  because  of  the  place  value, 
it  would  be  natural  to  extend  the  number  scale  to  the  right 
of  the  unit.  Such  an  extension  of  the  scale  would  give  rise 
to  the  decimal  fraction.  Consider  the  number  111.  The 
figure  at  the  right  indicates,  by  virtue  of  place  value,  one- 
tenth  of  the  value  indicated  by  the  figure  immediately  to 
its  left,  and  0.01  of  that  indicated  by  the  second  figure  to 
its  left.  If  this  same  scale  is  extended  to  the  right  of  the 
units  figure  we  have  the  decimal  fraction.  If  decimals 
originated  in  this  manner  the  subject  is  intimately  related 
to  integers  and  only  indirectly  related  to  common  frac- 
tions. A  rational  explanation  of  the  processes  involved  in 
decimal  fraction  may  be  based  upon  either  of  these  theories. 
De  Morgan  asserts  that  a  table  of  compound  interest  first 
suggested  decimal  fractions. 

Decimal  fractions  were  introduced  so  gradually  that 
their  origin  cannot  be  ascribed  to  any  one  person.  About 
1585  a  book  was  written  by  Stevin  in  which  he  attempted 
to  show  their  great  practical  value.  He  says,  "It  is  cer- 
tain that  if  the  nature  of  man  remains  in  the  future  as  it 
is  now,  he  will  not  always  neglect  so  great  an  advantage. ' ' 
The  world  is  not  always  quick  to  take  advantage  of  a  great 
improvement,  as  was  evidenced  by  the  tardiness  in  the 
adoption  of  the  Gregorian  calendar  and  as  is  evidenced 
to-day  by  the  opposition  to  the  metric  system.  It  was  not 
until  a  century  and  a  half  after  the  time  of  Stevin  that 
decimal  fractions  were  generally  taught.  Some  of  the 
common  fractions  of  the  fifteenth  century  had  become  so 
unwieldy  that  a  change  was  imperative.  One  author  used 
the  fraction  34fff  srf ,  anc*  anotner  required  the  square  root 
of  2520^IVW£r£?iV.  Tne  time  was  ripe  for  a  new  nota- 
tion in  fractions. 


DECIMAL  FRACTIONS  197 

Symbolism  of  Decimal  Fractions 

Decimal  fractions  appeared  in  a  number  of  interesting 
forms  before  the  decimal  point  was  used.  It  is  instructive 
for  teachers  to  know  that  the  point  is  not  necessary  in 
writing  decimal  fractions.  Fractions  had  been  used 
for  many  years  before  the  decimal  point  appeared.  Stevin, 
who  was  the  first  man  to  write  a  valuable  treatise  011  the 
subject,  seemed  to  appreciate  the  significance  of  the  new 
fractions,  but  his  system  lacked  a  suitable  notation.  He 
did  not  use  the  decimal  point.  If  he  wished  to  express 
5.992  he  wrote  it  gjff.  He  used  a  zero  to  indicate  the 
units  figure  and  placed  a  "1"  over  the  first  decimal  figure. 
Various  devices  were  used  by  other  writers  to  express 
decimal  fractions.  The  following  methods  were  in  actual 
use  for  such  a  number  as  5.992 : 


0      1    11     111  111 


5992  5092  59923  5|992  5992,  5|992 
Pellos  (1492)  unwittingly  made  use  of  the  decimal  point. 
This  is  the  first  instance  known  in  which  it  was  used  in  a 
printed  work.  The  point  was  also  used  by  Jobst  Biirgi  in 
a  manuscript  of  1592,  and  by  Pitiscus  in  some  trigono- 
metric tables  in  1612.  Kepler  in  1616  used  both  the  deci- 
mal point  and  the  parenthesis  to  separate  the  integer  and 
the  decimal  fraction. 

The  awkward  symbolism  for  decimal  fractions  gradually 
disappeared  and  either  the  point  or  the  comma  came  to  be 
generally  used.  The  symbolism  is  not  settled  even  to-day. 
In  the  United  States  three  and  twenty-five  hundredths  is 
expressed  as  3.25.  In  England  it  is  expressed  as  3  -25,  while 
in  Germany,  France,  and  Italy  it  is  3,25  or  325.  A  mere 
space  to  indicate  the  separation  of  the  integral  and  deci- 
mal units  is  not  uncommon  in  print.  A  zero  is  sometimes 
written  to  the  left  of  the  decimal  point  to  call  attention  to 
the  point  more  quickly.  Thus  .7  may  be  written  0.7. 


198  HOW  TO  TEACH  ARITHMETIC 

It  is  important  that  the  teacher  should  see  that  by  any 
one  of  the  above  devices  a  decimal  fraction  may  be  ex- 
pressed as  truly  as  if  the  decimal  point  were  used.  All  of 
the  devices  have  a  common  aim,  and  that  is  to  indicate 
where  the  integral  part  of  the  number  ends  and  the  decimal 
part  begins.  Any  symbol  or  combination  of  symbols  that 
will  indicate  this  may  be  used  instead  of  the  decimal  point. 
Since  the  purpose  served  by  the  symbols  just  noted  is  to 
separate  the  two  kinds  of  units, — integral  and  decimal,— 
the  name  separatrix  is  given  to  any  symbol  that  serves 
this  purpose.  The  point  is  but  one  of  a  large  number  of 
symbols  that  might  be  used.  It  appeals  to  us  as  the  best 
separatrix,  but  it  was  not  extensively  used  until  the  first 
quarter  of  the  eighteenth  century.  "  Simple  as  they  now 
appear,  the  development  of  decimal  fractions  was  too  great 
for  any  one  mind  or  age.  The  idea  of  their  use  gradually 
dawned  upon  the  mind,  one  mathematician  taking  up  what 
another  had  timidly  begun,  added  an  idea  or  two,  until 
the  whole  subject  was  at  length  fully  conceived  and 
developed."1 

Reading  and  Writing  Decimal  Fractions 

After  the  conception  of  the  decimal  fraction  has  been 
made  clear,  considerable  practice  should  be  given  in  the 
reading  and  writing  of  decimals.  Teachers  will  find  this 
work  greatly  facilitated  if  they  require  the  pupils  to 
memorize  the  orders  of  the  decimal  scale  and  the  position 
of  each  order  with  reference  to  the  decimal  point.  For 
example,  the  pupil  should  know  that  thousandths  is  the 
third  decimal  order,  millionths  the  sixth,  hundred- 
thousandths  the  fifth,  etc.  If  asked  to  write  four  hundred 
twenty-three  millionths,  the  pupil  should  at  once  think 

i  Brooks,  "Philosophy  of  Arithmetic/' 


DECIMAL  FRACTIONS  199 

that  since  millionths  is  the  sixth  order,  and  since  there  are 
three  digits  in  four  hundred  twenty-three,  he  must  insert 
three  zeroes ;  he  therefore  begins  at  the  decimal  point  and 
writes  three  zeroes,  then  the  figures  4,  2,  and  3.  There  is 
an  advantage  in  being  able  to  write  decimal  fractions  from 
left  to  right  just  as  we  write  integers.  The  pupil  who 
writes  decimal  fractions  from  left  to  right  does  not  need 
to  count  the  digits  after  he  has  written  the  expression  in 
order  to  insure  the  correctness  of  his  work.  We  do  not 
count  the  digits  after  we  have  written  an  integer  to  insure 
correctness,  and  there  is  no  reason  why  we  should  do  so  in 
writing  decimal  fractions.  Such  work  as  the  following 
is  of  no  little  value :  How  would  you  write  twenty-four 
ten-thousandths  ?  Answer.  Decimal  point,  two  zeroes,  2,  4. 
When  pupils  are  required  to  write  decimal  fractions  from 
dictation,  the  enunciation  should  be  very  distinct  and  the 
pupil  should  image  the  decimal  completely  before  writing 
it ;  he  should  then  write  it  continuously  from  left  to  right. 

To  read  a  pure  decimal:  Read  as  in  whole  numbers, 
then  state  the  name  of  the  decimal  order  of  the  figure  at 
the  right.  Thus  .047  is  read  forty-seven  thousandths. 

To  read  mixed  decimals:  Read  the  integral  part,  then 
the  decimal  part,  joining  the  two  parts  by  and.  Thus 
253.047  is  read  two  hundred  fifty-three  and  forty-seven 
thousandths. 

Mixed  decimals  may  also  be  read  as  improper  fractions. 
Thus  2.47  may  be  read  as  two  hundred  forty-seven  hun- 
dredths.  It  is  a  good  exercise  to  read  decimal  fractions 
as  tenths,  hundredths,  etc.  Thus  253.047  read  as  tenths  is 
2530.47  tenths ;  as  hundredths  it  is  25304.7  hundredths ;  as 
tens  it  is  25.3047  tens. 

Pure  or  mixed  decimals  may  also  be  read  by  stating  the 
order  which  each  digit  occupies.  Thus  253.047  may  be 
read  as  two  hundreds,  five  tens,  three  units,  four  hun- 


200  HOW  TO  TEACH  AEITHMETIC 

dredths,  seven  thousandths.  Hundredths  is  often  read  as 
per  cent.  Thus  .72  may  be  read  72  per  cent.  3.75  may  be 
read  three  hundred  seventy-five  per  cent. 

There  are  certain  types  of  decimal  fractions  that  are 
frequently  read  incorrectly.  Illustrations  of  these  types 
follow : 

2400.0006  should  be  read,  twenty-four  hundred  and  six 
ten-thousandths. 

.2406  should  be  read,  twenty-four  hundred  six  ten- 
thousandths. 

.Of  should  be  read,  two-thirds  of  a  tenth,  or  two-thirds 
tenths. 

.OOf  should  be  read,  two-thirds  of  a  hundredth,  or  two- 
thirds  hundredths. 

2.0|  should  be  read,  two  and  two-thirds  of  a  tenth. 

0.2f  should  be  read,  two  and  two-thirds  tenths. 

2.2|  should  be  read,  two  and  two  and  two-thirds  tenths. 

Attention  should  be  directed  to  the  fact  that  in  reading 
mixed  decimals  the  word  and  should  be  used  after  reading 
the  integral  part. 

The  expression  .f  is  sometimes  incorrectly  used  for  .Of. 
.Of  is  two-fifths  of  a  tenth,  but  the  expression  .f  has  no 
meaning  whatsoever  in  our  system  of  notation.  This  may 
be  shown  by  the  following :  f  =  .40 ;  two-fifths  of  a  tenth 
equals  f  of  -^  =  ^2¥-.04  =  .Of.  It  therefore  appears  that  .f 
is  neither  f  of  a  unit  nor  f  of  a  tenth. 

Comparative  Values 

Pupils  should  be  able  to  recognize  comparative  value  in 
decimal  fractions.  Ask  the  pupils  how  much  length  is 
represented  by  each  figure  in  .546  miles?  How  mn.eh 
money  is  represented  by  each  figure  in  $.394?  Pupils 
should  realize  that  0.1  is  greater  than  .098  and  that  .01  is 


DECIMAL  FRACTIONS  201 

greater  than  .00998.  Ask  the  pupil  to  state  the  decimal 
fraction  of  two  digits  that  is  nearest  in  value  to  .5743  or 
to  .03875.  It  is  not  wise  to  emphasize  the  decimal  orders 
beyond  the  sixth,  and  most  of  the  emphasis  should  be 
placed  upon  the  first  four  orders. 

Annexing  Zeroes 

The  effect  of  annexing  zeroes  to  the  right  of  a  decimal 
fraction  should  be  understood.  Ask  the  pupil  to  write  .14  ; 
this  is  the  same  as  .1  +  .04.  If  he  should  write  a  3  to  the 
right  of  the  4,  the  expression  would  be  .143.  How  much 
has  been  .added?  Evidently  .003  has  been  added.  If  a  2 
had  been  written  instead  of  a  3,  evidently  .002  would  have 
been  added.  If  a  zero  had  been  written,  evidently  no 
thousands  would  have  been  added,  or,  in  other  words,  the 
value  of  the  original  expression  would  not  have  been 
changed.  By  numerous  illustration  such  as  this  the  truth 
should  be  taught. 

.  The  effect  of  annexing  zeroes  to  the  right  of  a  decimal 
fraction  may  be  explained  also  by  showing  that  the  annex- 
ing of  zeroes  is  equivalent  to  multiplying  both  numerator 
and  denominator  by  the  same  power  of  10.  For  example, 


The  thing  to  be  chiefly  emphasized  in  the  teaching  of 
decimal  fractions  is  the  four  fundamental  operations. 

Addition  and  Subtraction 

Addition  and  subtraction  offer  no  difficulties  except 
those  previously  met  in  the  same  processes  with  integers. 
Care  should  be  taken  that  units  of  the  same  order  are 
placed  in  the  same  vertical  column  so  that  the  decimal 
points  stand  in  a  vertical  line.  The  decimal  point  in  the 


202  HOW  TO  TEACH  ARITHMETIC 

result  is  then  placed  directly  under  the  line  of  decimal 
points. 

Illustration: 

17.326 

0.0437 

9.801 

0.42 

27.5907 

Multiplication  and  Division 

There  are  two  methods  of  explaining  the  location  of  the 
decimal  point  in  multiplication  and  division  of  decimal 
fractions.  These  two  methods  are  based  upon  the  two  con- 
ceptions of  the  origin  of  the  fractions.  By  one  of  these 
methods  the  position  of  the  decimal  point  in  the  result  is 
determined  by  principles  of  common  fractions;  by  the 
other  it  is  determined  from  the  pure  decimal  conception 
of  the  number  system. 

By  the  first  method  the  decimal  fractions  are  reduced  to 
common  fractions  and  the  multiplication,  or  division,  is 
then  performed.  The  result  is  then  written  as  a  decimal 
fraction.  Multiplication  will  first  be  considered. 

Illustrations: 

.3x    .15=  W  x  TVo  =   iflhr   =-045 

.04  x.  007  =  T^xTo7^  =  TTHHhnr  =  .00028 

4.27  x.  005  =  fttxTT/W  =  ifHfa  =  -02135 


This  method  is  not  the  one  used  by  most  text-book  writers 
and  teachers,  but  it  is  of  value.  If  a  teacher  believes  that 
decimal  fractions  are  a  transition  from  common  fractions, 
and  if  he  wishes  his  method  to  be  in  harmony  with  the 
manner  in  which  the  race  developed  the  subject,  —  this  is 
not  always  desirable,  —  he  should  use  the  above  method. 

The  second  method  is  based  upon  the  theory  that  decimal 


DECIMAL  FRACTIONS  203 

fractions  originated  from  an  extension  of  the  number  scale 
to  the  right  of  the  units  order.  Before  taking  up  the 
explanation  by  this  method  the  effect  of  moving  the  decimal 
point  a  given  number  of  places  to  the  right  or  the  left 
should  be  thoroughly  understood.  Ask  the  pupil  to  com- 
pare the  values  of  the  following  numbers:  100,  10.0,  1.00, 
0.1,  and  to  state  the  effect  of  moving  the  decimal  point 
to  the  left.  Apply  the  principle  to  the  following  and  see 
if  it  is  true  :  428,  42.8,  4.28,  .428,  .0428.  Similarly,  teach 
the  effect  of  moving  the  decimal  point  to  the  right.  The 
pupil  should  also  be  led  to  discover,  or  review,  the  prin- 
ciple that  when  one  of  two  factors  is  multiplied  by  any 
number  whatsoever  and  the  other  is  divided  by  this  same 
number,  the  product  of  the  two  factors  is  not  changed. 
The  truth  of  this  principle  may  be  made  clear  by  illus- 
trations. 

Illustrations: 

7x4=^x4x10=28 


In  the  first  illustration  above  we  introduced  ten  as  a 
factor  in  the  numerator  and  also  in  the  denominator; 
hence  we  introduce  J--JJ-,  or  1. 

If  the  pupil  is  required  to  find  the  product  of  4  x  .07 
he  may  use  the  same  principle  as  when  he  is  required  to 
find  the  product  of  4  x  $7  or  4x7  books.  He  knows  that 
4x7  hundredths  is  28  hundredths,  or  he  may  write  it  as  .28. 
If  the  example  requires  the  product  of  .4  x  .07  he  may  use 
the  principles  referred  to  above.  He  may  multiply  the  .4 
by  10  and  divide  the  .07  by  10  without  changing  the  value 
of  the  product.  Hence,  4  x  .07  =  4  x  .007  =  .028.  Similarly, 
.03x.005  =  3x.00005  =  00015.  (In  this  case  one  factor  is 
multiplied  by  100  and  the  other  factor  is  divided  by  100.) 
Similarly,  32.54  x.005  =  3254  x.00005  =.16270. 


204  HOW  TO  TEACH  AEITHMETIC 

After  the  pupil  has  worked  several  examples  he  should, 
under  the  direction  of  the  teacher,  formulate  his  own  rule 
for  use,  and  he  should  see  that  whatever  operation  is 
necessary  to  make  the  multiplier  an  integer  should  be  per- 
formed upon  it,  and  the  inverse  operation  should  be 
performed  upon  the  multiplicand.  If  the  multiplier  is 
multiplied  by  10,  100,  or  1000,  the  multiplicand  should  be 
divided  by  10,  100,  or  1000,  respectively,  in  order  that  the 
value  of  the  product  may  be  unchanged. 

Multiplication  of  decimals  may  be  explained  also  in  the 
following  manner.  Required  to  multiply  .043  by  .05.  If 
we  multiply  .043  by  5,  the  result  is  .215,  but  this  example 
differs  from  the  given  example  only  in  that  the  multi- 
plier is  100  times  as  large  as  the  given  multiplier  (since 
5  =  100  x. 05).  Therefore  the  product  is  100  times  too 
large,  and  the  correct  product  is  y^  of  .215,  or  .00215. 

Whatever  method  is  used  should  appear  to  the  pupil  as 
rational,  and  after  a  sufficient  number  of  examples  have 
been  worked  he  should  formulate  the  rule  that  in  order 
to  multiply  two  decimal  fractions  we  proceed  as  in  multi- 
plication of  integers  and  then  point  off  in  the  product  as 
many  decimal  places  as  there  are  in  both  the  multiplicand 
and  the  multiplier.  If  the  product  does  not  contain  as 
many  figures  as  there  are  in  the  multiplicand  and  multiplier 
together,  the  necessary  zeros  must  be  prefixed  to  the 
product. 

After  the  rule  for  multiplication  has  been  rationally 
developed,  the  pupil  should  use  it  instead  of  developing 
the  principle  each  time  he  wishes  to  find  the  product  of 
two  decimal  fractions.  The  first  essential  is  to  make  the 
process  seem  reasonable  to  the  pupil,  and  the  second  is  to 
acquire  facility  in  the  application  of  the  process. 

Division  of  decimals  may  be  explained  by  first  reducing 
the  decimal  fractions  to  common  fractions  and,  after  per- 


DECIMAL  FRACTIONS  205 

forming  the  division,  expressing  the  quotient  by  decimal 
notation. 

Illustrations: 


.0045  *  .00005  =  Trf      -  =  90 


After  working  several  examples,  the  rule  which  ex- 
presses the  relation  between  the  number  of  decimal  places 
in  dividend,  divisor,  and  quotient  should  be  stated. 

Most  authors  and  teachers  explain  division  of  decimal 
fractions  by  use  of  the  principle  that  multiplying  or  divid- 
ing both  dividend  and  divisor  by  the  same  number  does 
not  change  the  value  of  the  quotient.  This  principle  is 
readily  understood  from  a  few  illustrations.  For  example, 
the  quotient  of  30^-  6Js  not  changed  if  both  the  30  and  the 
6  are  multiplied  or  divided  by  the  same  number. 

30-r6=(7x30)-s-(7x6)  or  30-3-6  =(i  of  30)-=-(i  of  6) 

If  the  pupil  is  required  to  work  the  example  .06  -=-  3,  he 
knows  the  quotient  just  as  he  knows  the  quotient  if  asked 
to  divide  $6  or  6  books  by  3.  If  required  to  divide  .06  by  .3 
he  may  use  the  principle  above.  Thus  .06  -r  .3  =  .6  -r  3  -  .2. 
(Both  dividend  and  divisor  were  multiplied  by  10.)  Simi- 
larly, .00015  -r  .005  =  .15  -r  5  =  .03.  (  Both  dividend  and  divisor 
were  multiplied  by  1000.) 

The  pupil  should  be  led  to  see  that  we  first  multiply  both 
dividend  and  divisor  by  whatever  will  make  the  divisor  an 
integer,  and  then  perform  the  division. 

After  a  sufficient  number  of  examples  have  been  worked 
by  the  above  method,  the  pupil  should  formulate  a  rule 
for  division  of  decimal  fractions.  Divide  as  in  integers 
and  point  off  in  the  quotient  a  number  of  decimal  places 
equal  to  the  excess  of  the  number  of  decimal  places  in  the 
dividend  over  the  number  of  decimal  places  in  the  divisor. 


206  HOW  TO  TEACH  AEITHMETIC 

This  rule  may  also  be  deduced  from  the  fact  that  divi- 
sion is  the  inverse  of  multiplication,  and  the  dividend  is 
the  product  of  the  divisor  and  quotient.  Since  the  number 
of  decimal  places  in  the  product  equals  the  number  in 
both  multiplicand  and  multiplier,  the  number  of  decimal 
places  in  the  quotient  must  equal  the  number  in  the 
dividend  minus  the  number  in  the  divisor. 

In  division  of  decimal  fractions  the  quotient  should  be 
written  above  the  dividend  in  such  a  manner  that  the 
decimal  point  of  the  quotient  will  be  directly  over  the 
decimal  point  of  the  dividend. 

Thus: 

5.483 
8)43.864 

* 

Indicated  multiplication  and  division  containing  deci- 
mals should  generally  be  rewritten  with  the  divisors 
changed  to  integers. 

Thus 

.47  x  12.25  x  1.03^  47  x!225  x  1.03 
.23x2.05x6  23x205x6 

(by    multiplying    both    numerator    and    denominator    by 
100x100). 

Contracted  Methods 

In  certain  work  in  science  and  in  actual  business  activi- 
ties operations  are  performed  involving  long  decimal 
fractions,  when  the  result  is  'desired  to  only  a  few  decimal 
places.  Perfect  measurements  are  rarely  possible  in  sci- 
ence, and  results  beyond  3  decimal  places  are  seldom 
required  in  business. 

If  a  measurement  is  correct  to  only  3  decimal  places, 


DECIMAL  FRACTIONS  207 

any  computation  in  which  it  is  directly  involved  can  not 
be  correct  to  more  than  3  decimal  places.  Much  unneces- 
sary labor  is  expended  in  some  computations  in  seeking  a 
product  to  more  than  3  decimal  places.  In  a  commercial 
transaction  such  a  result  as  $43.492736  would  be  considered 
as  $43.49.  It  is  always  a  waste  of  time  to  carry  out  results 
to  a  greater  degree  of  accuracy  than  the  data  upon  which 
the  results  are  based  or  the  situation  demands. 

The  processes  of  multiplication  and  division  involving 
decimal  fractions  may  frequently  be  abridged  by  eliminat- 
ing all  unnecessary  work.  There  is  a  real  advantage  in 
multiplication  in  beginning  with  the  highest  instead  of  the 
lowest  order  of  units  of  the  multiplier. 

Suppose  the  product  of  23.4271  by  1.3723  is  required 
correct  to  3  decimal  places.  In  order  to  be  sure  that  the 
result  is  correct  to  .001  it  is  wise  to  carry  the  partial 
products  to  .0001. 

Full  Form  Contracted  Method 

23.4271  23.4271 

1.3723  1.3723 

23.4271  23.4271 

7.02813  7.0281 

1.639897  1.6399 

.0468542  .0469 

.00702813  .0070 

32.14900933  32.1490 

Each  partial  product  is  increased  by  as  many  units  as 
would  have  been  carried  from  the  rejected  part:  0.5  is 
usually  regarded  as  one  to  be  carried. 

Approximations  are  often  desirable  in  division.  The 
following  illustration  will  indicate  the  method:1 

i  Beman  and  Smith,  * '  Higher  Arithmetic, ' '  p.  11. 


208  HOW  TO  TEACH  AKITHMETIC 

(1)     31416)329201(10.48 

3142  =  approximately  10  x  3141  (  6  ) 

150 

126  approximately  0.4  x  314  (16) 

24 
24  approximately  0.08x31  (416) 

In  contracted  multiplication  and  division,  considerable 
practice  is  necessary  to  give  one  confidence  and  to  decide 
correctly  what  number  should  be  carried.  Contracted 
methods  are  not  as  generally  used  in  the  schools  as  their 
importance  would  justify.  The  European  countries  use 
them  quite  extensively.  It  is  not  desirable  to  emphasize 
these  contracted  methods  in  all  communities,  but  in  cer- 
tain industrial  centers  some  attention  should  be  given  to 
them. 

Reduction  of  Decimal  to  Common  Fractions 

Decimal  fractions  are  easily  reduced  to  equivalent  com- 
mon fractions.  To  reduce  a  decimal  fraction  to  a  common 
fraction,  omit  the  decimal  point,  write  the  denominator 
of  the  decimal,  and  then  reduce  the  common  fraction  to 
its  lowest  terms. 

Thus    .08  =3^  =  -&• 

This  exercise  will,  in  the  case  of  some  decimals,  afford  a 
review  of  complex  fractions.    Thus 
gi        j  o 


02~l—     -  'I 

"1000"  2000  ~T:(nr 

Reduction  of  Common  to  Decimal  Fractions 

The  reduction  of  a  common  fraction  to  a  decimal  frac- 
tion may  be  explained  in  two  ways. 


DECIMAL  FRACTIONS  209 

First  Method.  A  common  fraction  may  be  regarded  as 
an  indicated  division,  f  may  be  regarded  as  2-f-5.  Since 
we  cannot  take  i  of  2  units  exactly,  we  reduce  the  2  units 
to  the  next  lower  denomination,  thus  making  20  tenths 
(2.0).  i  of  20  tenths  is  4  tenths  (.4).  Therefore  %  =  A. 

Reduce  f  to  a  decimal  fraction.  Analysis,  f  of  1  unit 
equals  y  of  2  units.  We  cannot  take  y  of  2  units,  so  we 
reduce  the  2  units  to  lower  denomination  and  get  20  tenths, 
or  200  hundredths,  or  2,000  thousandths,  etc.  \  of  2,000 
thousandths  equals  285  f  thousandths,  or  .285f. 

This  process  may  be  briefly  performed  by  placing  a 
decimal  point  after  the  numerator  and  dividing  by  the 
denominator. 

Second  Method.    Perform  any  operation  upon  the  frac- 
tion that  will  make  the  denominator  a  power  of  10  but  will 
not  change  the  value  of  the  fraction,  then  omit  the  denomi- 
nator and  express  it  by  the  position  of  the  decimal  point. 
Thus 

3xl25      375 
"" 


x1^5     1000 

4x8       32 


125x8     1000 
(1) 


" 


7        7x625,4375 
T*     16x625     10000 


The  first  of  these  methods  is  easier  for  the  pupil  in  most 
cases.  The  second  method  is  often  briefer  but  it  requires 
more  skill. 

A  common  fraction  cannot  always  be  reduced  to  a  pure 
decimal.  Thus  J  =  0.3  J  =  0.33J  =  0.333^.  No  common  frac- 
tion can  be  reduced  to  a  pure  decimal  if  the  denominator, 


210  HOW  TO  TEACH  AEITHMETIC 

when  the  fraction  is  reduced  to  its  lowest  terms,  contains 
other  prime  factors  than  2  and  5.  This  is  true  since  ten 
is  the  product  of  2  and  5  and  any  power  of  10  is  the 
product  of  an  equal  number  of  2s  and  5s.  In  reducing 
such  common  fractions  as  -J  or  \  to  decimal  fractions  the 
division  may  be  carried  as  far  as  is  necessary  and  then  the 
common  fraction  or  the  plus  sign  may  be  written.  Thus  \  - 
.14285f,  or  .14285  +. 

Circulating  Decimals 

The  reduction  of  certain  common  fractions  to  decimal 
fractions  lead  to  the  discovery  of  circulating  decimals. 
Circulates  are  an  outgrowth  of  the  Arabic  system  of  nota- 
tion. The  subject  is  very  interesting,  but  it  is  more  prop- 
erly treated  under  infinite  series  in  algebra  than  in  arith- 
metic, and  its  discussion  is  omitted  here  for  that  reason.1 

Aliquot  Parts 

The  pupil  should  know  the  aliquot  parts  of  10,  100,  and 
1000  which  are  frequently  used,  and  should  learn  their 
decimal  equivalents.  The  equivalents  for  £,  £,  J,  J,  £,  4,  J,  -J-, 
i*tr?  T\  !*£>  and  for  f ,  f ,  f ,  J,  f  should  be  thoroughly  mas- 
tered. 

Sequence  of  Common  and  Decimal  Fractions 

Decimal  fractions  are  taught  in  most  schools  of  the 
United  States  in  the  fifth  or  sixth  grades.  Especial  emphasis 
is  usually  put  upon  the  subject  in  the  6th  grade  and  per- 
centage furnishes  an  opportunity  to  review  decimals  in 
the  seventh  or  eighth  grades.  The  sequence  of  common 

i  Teachers  who  are  interested  in  the  subject  of  circulating  decimals 
Avill  find  it  treated  in  the  following  references:  Brooks,  "Philosophy 
of  Arithmetic/7  pp.  460-485;  McLellan  and  Dewey,  "The  Psychology 
of  Number/'  pp.  267-271;  Showalter,  "Arithmetical  Solution  Book/' 
pp.  49-55;  Wentworth,  "Practical  Arithmetic/7  pp.  354-356, 


DECIMAL  FRACTIONS  211 

and  decimal  fractions  is  not  yet  settled.  The  general  prac- 
tice in  the  United  States  is  to  teach  the  simpler  common 
fractions,  such  as  ^,  ^,  and  J  in  the  very  early  grades  and 
to  treat  the  entire  subject  of  common  fractions  more  or  less 
formally  in  the  fifth  grade.  Decimal  fractions  are  rarely 
taught  except  as  they  are  involved  in  the  writing  of  United 
States  money,  until  the  fifth  or  sixth  grade.  No  one  seri- 
ously advocates  the  complete  development  of  common  frac- 
tions before  any  consideration  is  given  to  the  subject  of 
decimal  fractions.  If  decimal  fractions  are  regarded  as 
a  different  notation  for  certain  common  fractions,  the  sub- 
ject may  be  taught  simultaneously  with  common  fractions. 
The  argument  usually  advanced  by  those  who  advocate 
the  teaching  of  decimal  fractions  before  common  fractions 
is  that  the  decimal  fraction  logically  comes  first  since  it  is 
a  natural  outgrowth  of  our  Arabic  system  of  notation. 
Those  who  believe  that  common  fractions  should  be 
taught  first  maintain  that  this  sequence  is  in  harmony 
with  the  historical  development  of  the  subjects  and  that 
the  concept  of  the  common  fractions  is  simpler  than  that 
of  the  decimal  fractions.  It  is  further  urged  that  pupils 
do  not  grasp  the  importance  of  decimal  fractions  and  the 
simplifications  which  they  effect  until  they  have  worked 
with  common  fractions  with  unlike  denominators. 

Problems  Involving  Decimals 

All  of  the  problems  of  decimal  fractions,  as  of  all  other 
subjects,  should  be  as  practical  as  possible.  They  should 
have  a  direct  and  intimate  relation  to  the  daily  activities 
in  which  decimal  fractions  are  so  extensively  used  and 
which  come  within  the  experience  of  the  pupil. 

Problems  involving  money,  distance,  speed  and  time; 
rainfall,  temperature,  crop  yields,  etc.,  are  often  of  inter- 


212  HOW  TO  TEACH  ARITHMETIC 

est  to  pupils  and  furnish  a  good  opportunity  for  computa- 
tion in  decimal  fractions. 

It  has  been  asserted  that  the  history  of  arithmetic  has 
been  a  slow  but  well  marked  growth  towards  the  decimal 
idea.  The  general  adoption  of  decimal  fractions  has  had  a 
very  marked  effect  upon  several  other  topics  of  arithmetic. 
Decimal  fractions  have  lessened  the  importance  of  the  sub- 
jects of  greatest  common  divisor  and  least  common  mul- 
tiple and  have  simplified  many  of  the  calculations  of 
science. 

The  importance  of  decimal  fractions  is  recognized  more 
today  than  ever  before  and  the  subject  is  one  of  the  most 
important  in  arithmetic. 


CHAPTER  XV 
PERCENTAGE 

Percentage  a  Language  Lesson 

The  subject  of  percentage  is  almost  wholly  a  language 
lesson.  No  new  mathematical  principles  are  involved  in 
any  of  its  applications.  When  the  child  knows  one-fourth, 
he  knows  twenty-five  per  cent,  all  but  the  name.  The  prob- 
lem is  to  teach  the  pupil  to  think  familiar  ideas  in  a  new 
language.  Pupils  should  not  feel  that  a  change  in  termi- 
nology involves  new  mathematical  processes.  Some  text- 
books on  arithmetic  treat  the  subject  of  percentage  as  if 
it  involved  new  mathematical  principles.  In  such  texts 
the  subject  is  treated  under  various  cases,  rules  and  for- 
mulas. The  best  teachers  of  arithmetic  to-day  do  not 
present  the  subject  in  this  way.  McClellan  and  Dewey 
state  that  the  teaching  of  percentage  by  cases,  rules  and 
formulas  is  a  mistake  from  both  the  theoretical  and  the 
practical  points  of  view.  It  is  a  mistake  on  the  theoretical 
side,  "because  it  asserts  or  assumes  a  new  phase  in  the 
development  of  number;  on  the  practical  side,  because  it 
substitutes  a  system  of  mechanical  rules  for  the  intelligent 
application  of  a  few  simple  principles  with  which  the  stu- 
dent is  perfectly  familiar."1  In  actual  business  prac- 
tice problems  are  not  classified  under  their  rules  and  cases. 
The  terms  base,  rate,  and  percentage  may  be  made  to  serve 
somewhat  the  same  purpose  in  this  subject  that  the  terms 

'McClellan  and  Dewey,  "The  Psychology  of  Number, "  p.  279. 

213 


214  HOW  TO  TEACH  AEITHMET1C 

numerator  and  denominator  serve  in  ^the  study  of  common 
fractions.  The  word  rate  may  also  properly  be  used  when 
it  refers  to  "rate  of  gain,"  "rate  of  interest/'  etc. 

Relation  betiveen  Percentage  and  Fractions 

Percentage  is  a  continuation  of  fractions.  It  is  a  special 
case  of  the  subject  and  it  may  be  made  to  afford  excellent 
practice  in  enlarging  the  ideas  of  fractions  and  in  securing 
greater  facility  in  using  them.  The  following  illustration 
will  serve  to  show  the  language  change  in  the  transition 
from  fractions  to  percentage.  A  man  had  700  chickens 
and  sold  one  out  of  every  two,  or  one  out  of  every  ten, 
or  one  out  of  every  50,  or  one  out  of  every  100,  or  one  out 
of  every  350,  how  many  were  sold?  The  above  problem 
would  be  classified  under  the  subject  of  fractions.  Since 
one  out  of  two  was  sold,  the  number  sold  was  i  of  700; 
or  since  1  out  of  10  was  sold  the  number  sold  was  -^  of 
700;  or  since  1  out  of  100  was  sold  the  number  sold  was 
T-JTT  of  700. 

If  we  agree  that  the  word  "per"  shall  mean  "out 
of"  when  used  in  this  connection,  the  above  state- 
ments would  be:  1  per  2,  or  1  per  10,  or  1  per  100.  If 
we  substitute  the  Latin  word  "decem"  for  ten  our  state- 
ment becomes  1  per  decem;  if  we  substitute  the  Latin 
word  "centum"  for  hundred,  our  statement  becomes  1  per 
centum.  If  we  abbreviate  the  word  centum  by  cutting  off 
the  last  two  letters  we  have  1  per  cent.  We  frequently 
abbreviate  words  in  this  way  in  arithmetic;  for  example, 
we  write  "int."  for  "interest"  and  "fract."  for  "frac- 
tion." One  per  cent,  therefore,  means  1  out  of  every  100. 
If  a  problem  involving  1  out  of  every  100 .  is  properly 
classified  as  a  problem  in  fractions,  it  is  certain  that  a 
problem  involving  one  per  cent  is  also  a  problem  in  com- 


PERCENTAGE  215 

mon  fractions.  Smith  states,  in  his  Kara  Arithmetical 
that  in  certain  old  manuscripts  %  used  to  be  written 
:*per  100."  It  became  $  about  1650,  and  this  became 
0/0,  then  %.  Percentage  came  in  as  a  separate  topic 
about  the  beginning  of  the  19th  century. 

The  mere  fact  that  a  quantity  is  measured  off  into  hun- 
dredths  instead  of  into  any  other  possible  number  of  parts 
appears  to  be  no  valid  reason  for  considering  percentage 
as  a  new  phase  in  the  development  of  number.  Is  the  proc- 
ess of  computing  by  hundredths  "to  be  broadly  distin- 
guished as  a  mental  operation  from  a  process  of  comput- 
ing by  eighths,  or  tenths,  or  twentieths,  or  fiftieths?  If 
the  difference  between  fractions  and  percentage  is  not  a 
difference  in  logical  or  psychological  processes,  but  chiefly 
a  difference  in  handling  number  symbols,  is  it  worth  while 
to  invest  the  subject  with  an  air  of  mystery  and  to  invent, 
for  the  edification  of  the  pupil,  from  six  to  nine  'cases' 
with  their  corresponding  rules  and  formulas  ?"2 

Applications 

Percentage  admits  of  applications  in  many  fields,  and 
the  increasing  use  of  decimal  fractions  is  increasing  the 
number  of  problems  to  which  percentage  is  applied.  In  the 
texts  of  a  few  decades  ago  almost  all  of  the  problems  in 
percentage  had  a  financial  basis,  but  to-day  the  best  texts 
contain  numerous  problems  involving  percentage  when 
money  considerations  are  in  no  way  involved.  Some  of 
the  applications  are  difficult  to  teach  because  the  transac- 
tions are  foreign  to  the  experience  of  most  of  the  pupils. 
One  of  the  chief  difficulties  which  pupils  encounter  in 

i  Smith's  "Kara  Arithmetica, "  p.  440. 

The  symbol  %,  read  "per  mil"  and  meaning  "by  the  thou- 
sandths/' is  somewhat  used  in  Europe,  but  is  rarely  used  in  the 
United  States. 

sMcClellan  and  Dewey,  "The  Psychology  of  Numbers/'  p.  280. 


216  HOW  TO  TEACH  AEITHMETIC 

studying  the  applications  of  percentage  is  the  fact  that  they 
do  not  get  a  clear  comprehension  of  the  terms  used.  Un- 
less the  pupil  has  a  clear  conception  of  the  terms  employed 
he  cannot  form  accurate  judgments  and  conclusions.  Too 
often  the  terms  used  are  but  "words,  words,  words,"  to 
the  pupil.  The  various  applications  of  percentage  should 
be  taught  by  correlating  them  as  closely  as  possible  with 
the  actual  situations  in  which  they  are  used. 

Anything  that  increases  the  interest  of  the  pupil  and 
helps  him  to  associate  the  terms  and  the  problems  with  ac- 
tual situations  in  which  they  occur  is  worth  while  unless 
an  excessive  amount  of  time  is  expended.  Any  material 
that  will  vitalize  the  work  may  be  used  to  advantage  by 
the  teacher.  The  pupil  observes  numerous  business  cus- 
toms and  procedures  in  his  daily  activities  and  his  obser- 
vations should  be  encouraged  and  made  as  accurate  as 
possible. 

It  is  important  that  the  pupil  should  have  an  under- 
standing of  his  natural  environment  and  it  is  important 
that  he  should  have  an  understanding  of  his  business  en- 
vironment. An  understanding  of  both  natural  and  business 
environment  is  essential  for  the  life  work  of  a  successful 
individual. 

Whatever  problems  are  used  in  percentage  should  be 
in  harmony  with  actual  business,  industrial  and  scientific 
practice.  All  problems  should  be  adjusted  to  present-day 
activities.  Dewey's  criticism  that  much  that  is  learned  in 
school  cannot  be  applied  in  daily  life  is  especially  perti- 
nent here.  "The  fact  that  the  arithmetic  of  business  is 
the  arithmetic  of  common  sense  should  not  for  a  moment 
be  lost  sight  of  in  drilling  classes  in  this  branch  of  our 
schools."1  The  teacher  should  seek  to  train  the  pupil  to 

i  N.  E.  A.  Committee  Report  on  Business  College  Course,  Vol.  II., 
p.  2,163. 


PERCENTAGE  217 

grasp  conditions  and  to  develop  the  power  to  apply  the 
processes  to  actual  business  and  industrial  situations.  All 
applications  of  percentage  not  consistent  with  modern  prac- 
tice should  be  omitted.  One  of  the  chief  objects  should  be 
to  develop  commercial  and  industrial  efficiency  and  social 
insight. 

Equivalents  of  Certain  Common  Fractions 

The  teacher  should  impress  upon  the  mind  of  the  pupil 
the  fact  that  "per  cent"  is  identical  in  meaning  with 
6  1  hundredths.  '  '  Six  per  cent  of  a  quantity  equals  six 
one-hundredths  of  it.  10%=^.=  10;  100%=H$  =  1.00; 
357  %=fH=  3-57;  f%  =f/100  =.00375. 

Many  problems  involving  the  reduction  of  common  frac- 
tions to  per  cent  and  of  per  cent  to  common  fractions  should 
be  solved.  Most  of  the  work  should  be  done  orally  and  the 
pupil  should  be  familiar  with  the  fractional  equivalents 
for  the  most  common  per  cents  of  business.  These  should 
be  memorized  and  drilled  upon.  The  following  are  com- 
monly used: 


6  =  4  25  %  =  i 

6f  %=A          1*1  •%=*  33496=1  66}  %=f 

81  %  -  A          16f  %  =  4  50  %  =4  83i  %  =| 

10    %=&          20     96  =  4  374%'  =  f  100     %  =  1 


The  equivalents  for  4*  IT  an^  iV  are  not  frequently  used. 

There  should  also  be  drill  in  expressing  decimal  frac- 
tions as  per  cent.  For  example,  ,1  =  10%;  .13-13%; 
.9  =  90%;  .004  =  fo%,  or  .4%;  ,0(X)5=TJt%,  or  0.05%; 
0.00625  =  f%;  .Ot)|  =  f%;  5.24  =  524%. 

The  pupil  should  see  that  the  reduction  of  a  decimal 
fraction  to  per  cent  involves  simply  the  reduction  of  the 
decimal  fraction  to  hundredths  and  then  instead  of  read- 


218  HOW  TO  TEACH  ARITHMETIC 

ing  it  hundredths  we  call  it  "per  cent/'  which  means 
hundredths. 

The  Three  Problems  of  Percentage 

In  percentage  and  its  application  there  are  but  three 
fundamental  mathematical  ideas  involved  and  all  of  these 
are  familiar  to  the  pupil  before  he  begins  the  formal  study 
of  the  subject.  The  most  complex  problems  in  percentage 
contain  no  new  mathematical  principles.  A  pupil  who  has 
mastered  the  four  fundamental  operations  with  integers, 
and  with  common  and  decimal  fractions,  and,  who  can  re- 
duce common  fractions  to  decimal  fractions  and  vice  versa, 
should  have  little  or  no  difficulty  with  percentage.  He 
may  be  unable  to  solve  the  problems,  but  his  failure  will  be 
due  to  a  failure  to  understand  the  terms  employed  and  to 
think  old  ideas  in  terms  of  new  symbols.  Not  the  mathe- 
matics, but  the  terminology  of  percentage  is  the  crux  of  the 
subject.  Since  only  three  fundamental  mathematical  ideas 
are  involved  in  the  problems  of  percentage  it  is  economical 
of  time  and  effort  for  the  teacher  to  emphasize  these  three 
ideas  before  proceeding  far  with  the  subject. 

The  first  fundamental  idea  is:  To  find  any  per  cent  of 
a  given  quantity.  Find  7%  of  45.  The  pupil  knows  that  per 
cent  means  "hundredths"  and  from  his  study  of  decimal 
fractions  he  knows  how  to  find  .07  of  a  given  quantity. 

7  %  of  45  =  .07  x  45  =  3.15 
32  %  of  372  =  .32  x  372  =  119.04 

To  find  any  per  cent  of  a  quantity,  multiply  the  given 
quantity  by  the  given  per  cent  expressed  as  hundredths. 
It  is  evident  that  a  per  cent  of  a  given  quantity  may  be 
regarded  as  the  product  of  two  factors. 

When  per  cents  greater  than  100%  are  involved  the 
meaning  "out  of  100 "  is  not  so  clearly  understood.  The 


PERCENTAGE  219 

pupil  will  easily  understand  the  meaning  of  200%,  how- 
ever, if  the  fact  that  %  means  hundredths  is  thoroughly 
understood. 

Find  f%  of  416.     f%  =  . 00375;   therefore  f%  of  416  = 

.00375x416  =  1.56.  Or  we  may  say,  t%  =  -^r=-dhr;  tnere- 
fore  f%  of  416=^  of  416  =  ^  of  52  =  j^  =  1.56. 

It  is  evident  that  the  application  of  this  principle  in- 
volves no  new  mathematical  knowledge.  The  symbols  in- 
volved are  new,  but  the  principle  is  not  new. 

The  Second  Fundamental  Idea  Is: 

Find  what  per  cent  one  number  is  of  another. 

Four  is  what  per  cent  of  eight  ? 

In  the  preceding  examples  we  were  required  to  find  a 
given  per  cent  of  a  number.  In  this  problem  we  are  told 
that  if  we  take  a  certain  per  cent  of  8  the  result  is  4. 

In  other  words,  we  have  given  the  product  of  two  factors, 
also  one  of  the  factors  to  find  the  other  factor.  This  in- 
volves no  new  principle.  If  we  divide  the  product,  (4) 
by  the  given  factor,  (8),  the  result  is  the  other  factor. 
Since  the  problem  asks  for  the  result  in  terms  of  per  cent 
(hundredths)  we  express  the  result  not  as  -|,  but  as  -f-fo 
or  50%. 

Similarly : 

7  is  what  %  of  17? 

?  x  17  =  7.  The  unknown  factor  is  found  to  be  T\,  or 
.41W  or  41TV%. 

The  Third  Fundamental  Idea  Is: 

Find  a  number  when  a  certain  per  cent  of  it  is  known. 
40%  of  a  number  =  10.    Find  the  number. 


220  HOW  TO  TEACH  ARITHMETIC 

Ten  is  the  product  of  two  factors,  one  of  which,  .40,  is 
known.  To  find  the  other  factor,  divide  the  product,  (10) 
by  the  known  factor,  .40,  the  result  (25)  is,  therefore, 
the  other  factor,  or  the  required  number. 

.40x1=10 

Some  teachers  prefer  to  explain  these  processes  by  the 
use  of  unitary  analysis.  The  above  explanations,  how- 
ever, are  brief  and  are  easily  understood  by  the  pupils, 
furthermore  they  involve  no  new  mathematical  ideas  and 
are  serviceable. 

The  Solution  of  Problems 

The  pupil  who  has  a  thorough  mastery  of  the  preced- 
ing principles  does  not  need  to  solve  the  problems  of  per- 
centage by  rules  and  cases.  Sometimes  he  should  use  the 
purely  fractional  form,  at  other  times  the  percentage  form 
and  at  still  other  times  he  will  combine  the  fractional  and 
the  percentage  forms.  "Practice  in  the  application  of  the 
principles  should  enable  the  pupil  to  use  all  forms  with 
equal  facility  and  to  determine  in  any  given  problem  which 
of  the  forms  will  lead  to  the  most  elegant  and  concise  solu- 
tion." No  set  form  should  be  required  after  the  reasons 
for  the  various  processes  are  understood,  but  the  teacher 
should  require  the  pupil  to  express  himself  in  concise  and 
accurate  language.  Many  good  teachers  in  the  upper  gram- 
mar grades  require  the  solution  of  problems  in  percentage 
in  step  form  with  the  actual  work  of  the  elementary  opera- 
tions omitted.  Too  great  insistence  upon  minute  details 
in  written  work  may  disgust  and  discourage  some  pupils. 
The  important  thing  is  to  emphasize  the  relation  of  what 
is  given  to  what  is  required. 

If  the  problem  states  that  25%  of  a  number  is  18,  the 


PERCENTAGE  221 

pupil  should  immediately  see  that  the  number  is  4x18,  or 
72.  If  the  problem  requires  33^%  of  480  the  pupil  should 
take  ^  of  480.  Some  object  to  the  use  of  common  frac- 
tions in  problems  which  involve  percentage.  It  must  be 
remembered  that  percentage  is  studied  in  the  grades  be- 
cause of  its  practical  applications  and  the  pupil  should, 
whenever  possible,  solve  the  problems  of  percentage  in  the 
manner  in  which  these  problems  are  solved  in  actual  life 
outside  of  the  school.  No  business  man  would  solve  the 
following  problem  by  keeping  it  in  its  percentage  form. 
Twenty-five  per  cent  of  the  cost  of  an  article  is  $14,  find 
the  cost.  The  practical  man  would  immediately  find  the 
cost  by  finding  4x$14.  "The  arithmetic  of  business  is 
the  arithmetic  of  common  sense. " 

A  Source  of  Failure  in  Percentage 

One  of .  the  most  frequent  sources  of  error  in  the  study 
of  percentage  is  due  to  the  failure  of  the  pupil  to  keep 
clearly  in  mind  per  cent  of  what.  In  all  early  work  in  per- 
centage teachers  should  require  pupils  to  name  the  quan- 
tity upon  which  the  per  cent  is  based. 

Questions  such  as  the  following  will  help  to  impress  this 
important  point. 

1.  What  is  25%  of  40? 

2.  40  is  25%  of  what  number? 

3.  40  is  what  %  of  25  ? 

4.  40  is  25%  greater  than  what  number? 

5.  25  is  what  %  less  than '40? 

6.  What  number  is  25%  less  than  40? 

7.  40  is  25%  less  than  what  number? 

8.  40  is  what  %  greater  than  25  ? 

The  following  type  of  problem  is  especially  good  to  im- 
press the  importance  of  keeping  in  mind  the  quantity 


222  HOW  TO  TEACH  AEITHMETIC 

that  is  to  be  made  the  basis  in  a  problem  involving  per- 
centage. 

Water  expands  10%  of  its  volume  when  it  freezes.  What 
per  cent  of  its  volume  does  ice  contract  when  it  melts? 

The  artificial  problems  found  in  many  text-books  are  of 
the  same  general  type.  A's  money  is  25%  more  than  B's. 
B's  money  is  what  per  cent  of  A's? 

Solution : 

Since  A's  money  is  25%  of  B's  more  than  B's,  there- 
fore, A's  money  =  125%  of  B's  money;  therefore,  1%  of 
B's  money  =  T^  of  A's  money;  therefore,  100%  of  B's 
money  =  yff  °^  A's  money,  or  f,  or  80%  of  A's  money. 

The  problem  may  also  be  solved  by  using  common  frac- 
tions throughout. 

Types  of  Problems 

Numerous  problems  which  do  not  involve  money  should 
be  given  in  percentage.  The  percentage  of  increase  or  de- 
crease in  population  for  various  cities  from  1900  to  1910; 
the  percentage  of  games  won  or  lost  by  the  school  teams; 
the  percentage  of  attendance  at  school  on  a  given  day;  the 
average  heights  of  boys  and  of  girls  at  different  ages  and 
the  yearly  percentage  of  increase;  the  average  weight  and 
strength  at  different  ages;  the  per  cent  of  pupils  in  each 
grade  of  the  school  based  on  the  total  school  enrollment; 
the  percentage  of  boys  and  girls  in  a  given  grade;  the 
per  cent  of  waste  in  cutting  the  largest  possible  circle  out 
of  a  square ;  the  per  cent  of  decrease  when  various  articles 
are  dried  or  baked;  the  percentage  of  nutritive  matter  in 
various  food  articles ;  the  per  cent  of  yield  of  a  given  crop 
per  acre  as  compared  with  other  years ;  these  suggest  a  few 
of  the  many  problems  not  involving  money  that  may  be 
given  to  the  pupils. 


CHAPTER  XVI 

THE  APPLICATIONS  OF  PERCENTAGE 

PEOFIT   AND   LOSS 

The  subject  of  profit  and  loss  is  one  of  the  important 
applications  of  percentage.  It  is  closely  connected  with 
business  activities  and  the  problems  should  be  in  harmony 
with  business  practice.  The  experiences  of  the  pupils 
should  be  utilized  as  the  basis  for  problems  whenever  this 
is  possible,  especially  at  the  beginning  of  the  work.  The 
commercial  and  industrial  interests  of  the  community  should 
be  the  source  of  numerous  problems.  In  most  of  the  prob- 
lems involving  profit  and  loss  which  the  business  man  is 
called  upon  to  solve  the  cost  is  known.  Sometimes  a  mer- 
chant knows  a  possible  selling  price  and  must  decide  how 
much  he  can  afford  to  pay  to  make  a  certain  per  cent. 
Some  problems  involving  inverse  cases  may  be  properly 
introduced.  It  is  said  that  many  business  men  use  a  fixed 
selling  price  which  is,  for  example,  twenty-five  per  cent 
greater  than  the  cost.  When  this  is  done  the  business  man 
knows  at  once  that  his  profits  are  20%  of  his  receipts,  since 
a  gain  of  25%  on  the  cost  is  20%  of  the  selling  price. 
(25%  of  the  cost  is  20%  of  125%  of  the  cost). 

No  Rules  or  Cases  Necessary 

No  formal  rules  or  cases  are  necessary  or  desirable  in 
solving  the  problems  in  profit  and  loss.  The  teacher  should 
emphasize  the  fact  that  profit  and  loss  are  based  upon  the 

223 


224  HOW  TO  TEACH  ARITHMETIC 

cost  while  commercial  discount  is  based  upon  the  list 
price.  The  problems  in  profit  and  loss  may  be  grouped 
under  five  general  headings. 

I.  Given  the  cost  and  the  rate  per  cent  of  profit,  or  loss, 
to  find  the  profit,  or  loss,  and  the  selling  price. 

II.  Given  the  cost  and  the  profit,  or  loss,  to  find  the  rate 
per  cent  of  profit,  or  loss,  and  the  selling  price. 

III.  Given  the  profit,  or  loss,  and  the  rate  per  cent  of 
profit  or  loss,  to  find  the  cost. 

IV.  Given  the  selling  price  and  the  rate  per  cent  of  profit 
or  loss  to  find  the  cost. 

V.  Given  the  cost  and  the  selling  price  to  find  the  rate 
per  cent  of  profit  or  loss. 

Since  profit  and  loss  are  always  reckoned  on  the  cost  the 
procedure  should  be  suggested  by  the  nature  of  the 
problems. 

Many  problems  should  be  solved  without  the  use  of  paper 
or  pencil.  For  example: 

a.  The  cost  is  8  cents  and  the  selling  price  is  10  cents. 
Find  %  of  gain. 

b.  The  cost  is  12  cents  and  the  selling  price  is  12  cents. 
Find  %  of  loss. 

In  problem  "a"  the  pupil  should  see  at  once  that  the 
gain  is  2  cents  and  the  per  cent  gained  is  whatever  per 
cent  2  cents  (the  gain)  is  of  8  cents  (the  cost).  The  only 
mathematics  involved  in  such  problems  is  a  subtraction 
and  "to  find  what  per  cent  one  quantity  is  of  another 
quantity." 

Type  Solutions 

The  following  solutions  are  intended  to  be  merely  sug- 
gestive. It  is  not  assumed  that  they  are  the  only  correct 


THE  APPLICATIONS  OF  PERCENTAGE  225 

ones.     Teachers  and  pupils  are  cautioned  against  inac- 
curate expression  in  this  subject.1 

1.  Cost  =  $420.    Rate  of  gain  =  20%.    Find  gain  and  sell- 
ing price.    20%  (or  i)  of  $420  =  $84  =  gain. 

Selling  price  -  $420  +  $84  =  $504. 

It  is  evident  that  the  selling  price  might  have  been  de- 
termined by  .finding  120%  (or  I)  of  $420. 

2.  The  cost  is  $36.     The  loss  is  $4.     Find  selling  price 
and  per  cent  of  loss. 

The  selling  price  is  $36  -  $4  =  $32. 

The  per  cent  of  loss  is  whatever  per  cent  $4  (the  loss) 
is  of  $36  (the  cost). 

The  loss  is  |  or  11^%  of  the  cost. 

3.  The  gain  is  $81.     The  rate  of  gain  is  12£%.     Find 
cost  and  selling  price. 

12i%  (or  £)  of  cost  =  $81. 
Therefore  100%  (or  f)  of  cost  =  $648. 
Therefore  selling  price  =  $648  +  $81  =  $729. 

4.  The  selling  price  was  $560  and  the  rate  of  loss  20%. 
Find  the  loss  and  the  cost. 

Since  the  selling  price  was  20%  of  the  cost  less  than  the 
cost,  it  was  80%  of  the  cost. 

Therefore  80%c  of  the  cost  =  $560. 

Therefore  100  %  of  the  cost  =  $700. 

Therefore  the  loss  =  $140. 

The  above  problem  may  also  be  solved  as  follows: 


Let  C  =  the  COSt  The  introduction  of  the 

0  20c  -  the  loss  symbol  "  C  »  is  a  decided 

advantage    in    the    solu- 

0.80c  =  selling  price  =  $560  tion     of     such     problems 

<£K(?A  and     more     emphasis 

therefore       C=^  ^=$700  should  be  given  to  such 

.80  solutions    in    the   schools. 

i  See  chapter  on  Accuracy,  p.  50-51. 


226  HOW  TO  TEACH  ARITHMETIC 

5.  The  selling  price  was  $30 ;  the  cost  was  $18.  Find 
rate  per  cent  of  gain. 

The  gain  was  $30  -  $18  =  $12. 

The  rate  of  gain  is  whatever  per  cent  $12  (the  gain)  is 
of  $18  (the  cost).  $12  is  f,  or  66f%,  of  $18.  The  gain  is, 
therefore,  66f  %. 

In  every  problem  in  profit  and  loss  in  which  the  per 
cent  of  gain  or  loss  is  given,  three  per  cents  (or  fractions) 
may  always  be  found. 

The  first  of  these  is  always  known.  It  is  this,  100%  of 
the  cost  =  the  cost. 

The  second  of  these  is  always  given  in  the  problem. 

The  third  may  always  be  found  by  adding  or  subtracting 
the  first  two. 

For  example,  in  the  second  problem  above,  it  is  known 
that  100%  of  cost  =  cost.  . 

20%  of  cost  =  loss. 

80%  of  cost  =  selling  price. 

Instead  of  using  the  per  cent,  the  equivalent  fractions 
may  be  used. 

For  example :  The  selling  price  is  $390 ;  the  rate  of  gain 
is  30% ;  find  the  cost. 

|$  of  the  cost  =  the  cost.     (This  is  always  known.) 

T%  of  the  cost  =  the  gain.     (The  problem  states  this.) 
||  of  the  cost  =  selling  price.    (By  combining  the  first  two. ) 

Therefore, 

if  of  the  cost  =  $390. 

i£  of  the  cost  =  {f  of  $390  =  $300. 

The  above  problem  may  be  more  briefly  solved  as  follows : 

c  =  the  cost 
1.30c  =  $390 

C=  $^-  =$300.        The  cost  is  $300. 


THE  APPLICATIONS  OF  PERCENTAGE  227 

A  man  sold  two  lots  for  $150  each.  He  gained  25% 
on  the  first  and  lost  25%  on  the  second.  Find  the  entire 
gain  or  loss  by  the  transaction. 

Solution: 

125%  of  cost  of  first  lot  =  selling  price  of  first  lot.  (Since 
it  was  sold  at  a  gain  of  25%.) 

Therefore,  125%  of  cost  of  first  lot  =  $150. 

100%  of  cost  of  first  lot  =  $120. 

75%  of  cost  of  second  lot  =  $150.  (Since  it  was  sold  for 
$150  at  a  loss  of  25%.) 

100%  of  cost  of  second  lot  =  $200. 

Cost  of  both  lots  was  $320.  (  $120  +  $200.  )  Selling  price 
of  both  was  $300. 

The  loss  was  $20.  The  rate  of  loss  was  whatever  per 
cent  $20  (the  loss)  is  of  $320  (the  cost).  $20  is  6J%  of 
$320.  The  loss  was,  therefore,  6£%.  Such  problems  as 
this  frequently  confuse  pupils  because  they  do  not  keep  in 
mind  the  "per  cent  of  what." 

I  sold  |  of  an  article  for  f  of  the  cost  of  the  whole  article. 
Find  the  rate  per  cent  of  gain  or  loss. 

Since  f  of  the  article  sold  for  f  of  the  cost  of  the  whole, 
^  of  the  article  sold  for  ^  of  f,  or  f  of  the  cost  of  the  whole  ; 
therefore  f  of  the  article  sold  for  f  of  the  cost  of  the  whole. 

The  gain  was,  therefore,  £  of  the  cost,  or  12J%. 

The  above  solution  may  be  considerably  abridged  by 
using  "c"  for  cost  and  "s.  p."  for  selling  price. 

Thus:  fs.p.  =  fc. 
£  s.  p.  =  §c. 
fs.p.=98c. 


A  careful  study  of  the  illustrative  problems  in  profit 
and  loss  will  corroborate  the  statement  found  on  page  218 


228  HOW  TO  TEACH  AEITHMETIC 

in  regard  to  the  mathematics  involved  in  the  applications 
of  percentage. 

The  teacher  should  supplement  the  text-book  problems 
in  profit  and  loss  by  using  many  of  the  problems  given 
under  the  subject  of  fractions. 

COMMERCIAL   DISCOUNT 

The  subject  of  commercial  discount  is  usually  taught  in 
the  seventh  or  eighth  grade.  It  is  intimately  related  to 
business  activities.  Trade  or  Commercial  Discount  is  the 
deduction  from  the  list  price  of  an  article.  Sinc.e  such  dis- 
counts are  usually  made  "to  the  trade'7  the  term  trade 
discount  has  arisen.  .Discounts  are  usually  expressed  in 
per  cents  or  in  fractions.  The  trend  of  business  practice 
in  recent  years  has  been  towards  per  cents  that  are  easily 
reduced  to  common  fractions/  so  that  the  discount  can  be 
computed  easily.  Thus  a  discount  of  16f%  is  easier  to 
compute  than  one  of  15%.  Discounts  are  sometimes  quoted 
in  the  following  manner :  20  and  -J  off.  This  means  that 
a  discount  of  20%  is  taken  from  the  list  price  and  then  a 
discount  of  33;!%  (i)  ig  taken  from  this  remainder. 

Reasons  for  Discounts 

Among  the  most  important  reasons  for  granting  dis- 
counts are:  For  cash;  on  account  of  fluctuations  in  the 
market  price;  to  avoid  the  frequent  publication  of  large 
catalogues ;  for  large  amounts  purchased.  Wholesalers  and 
manufacturers  usually  publish  a  "list  price "  and  then 
allow  a  certain  per  cent  of  discount  "to  the  trade."  Goods 
listed  but  not  subject  to  discount  are  marked  "net."  A 
wholesale  house  usually  has  some  such  statements  as  the 
following  upon  its  bills:  Terms:  "4  months,  30  days  less 


THE  APPLICATIONS  OF  PERCENTAGE  229 

5%,"  or  "30  days,  2%  off  10  days,"  which  mean  that 
purchasers  are  entitled  to  a  credit  of  4  months,  but  will  be 
allowed  5%  discount  if  the  bills  are  paid  within  30  days; 
or  that  a  credit  of  30  days  is  allowed,  but  the  purchaser 
will  be  allowed  2%  discount  if  the  bills  are  paid  within 
10  days.  Sometimes  goods  are  paid  for  before  the  shipping 
date  for  the  goods  has  arrived.  This  is  called  "anticipat- 
ing a  bill."  A  discount  equivalent  to  the  current  rate  of 
interest  is  usually  allowed  on  such  a  transaction.  When 
possible,  teachers  should  consult  trade  journals  which  quote 
prices  and  discounts. 

Successive  Discounts 

When  more  than  one  discount  is  given,  the  first  dis- 
count is  reckoned  upon  the  list  price  and  the  others  are 
reckoned  upon  the  successive  remainders  after  the  pre- 
ceding discounts  have  been  deducted.  No  two  successive 
discounts  are  reckoned  upon  the  same  amount,  hence  suc- 
cessive discounts  can  never  be  added  to  find  the  equivalent 
single  discount.  Two  successive  discounts  of  20  and  10 
are  not  equivalent  to  a  single  discount  of  20  +  10.  In  all 
problems  involving  discounts  the  pupil  must  keep  in  mind 
the  basis  upon  which  the  discount  is  to  be  computed. 

Many  teachers  have  the  idea  that  successive  discounts 
must  be  removed  in  the  order  in  which  they  are  quoted. 
The  discounts  may  be  removed  in  any  order.  The  order 
does  not  affect  the  result.  Successive  discounts  of  20,  10, 
and  5  will  produce  the  same  net  price  whether  removed 
in  the  order  just  stated  or  in  any  of  the  following  orders : 
10,  5,  20,  or  20,  5,  10,  or  10,  20,  5,  or  5,  10,  20,  or  5,  20,  10. 
The  truth  of  this  statement  may  be  easily  verified. 

Assume  any  marked  price,  for  example,  $360. 

Suppose  the  discounts  are  quoted  at  20,  10,  and  5. 


230  HOW  TO  TEACH  AEITHMETIC 

First  Solution 

$360  =  list  price  or  marked  price 

.20 

$72.00,  first  discount 
$360  -$72  =  $288  =  basis  for  second  discount 

.10 

$28.80,  second  discount 
$288  -$28.80  =  $259.20  =  basis  for  third  discount 


_ 

$  12.96,  third  discount 
$259.20  -  $12.96  =  $246.24  -  net  price 

Second  Solution 

$360  =  list  price  or  marked  price 

.10 

$36.00,  first  discount 

$360  -  $36  =  $324  =  basis  for  second  discount 
.05 


$16.20,  second  discount 
$324  -$16.20  =  $307.80,  basis  for  third  discount 

.20^ 

$  61.56,  third  discount 
$307.80  -  $61.56  =  $246.24  =  net  price 

The  net  price  is  the  same  in  each  case. 

It  is  apparent  that  the  basis  for.  the  second  discount 
might  have  been  found  by  taking  80%  of  $360,  instead  of 
first  finding  20%  of  $360  and  subtracting  this  amount  from 
$360.  Similarly  the  basis  for  the  third  discount  might 
have  been  found  by  taking  90%  of  the  basis  for  the  sec- 
ond discount  and  the  net  price  might  have  been  found  by 
taking  95%  of  this  amount.  In  other  words,  we  first  find 
80%  of  the  marked  price,  then  90%  of  this  amount  and 


THE  APPLICATIONS  OF  PERCENTAGE  231 

finally  95%  of  this  amount.  The  above  operations  may 
be  indicated  as  follows: 

.80  x  .90  x  ,95  x  $360  =  $246.24.  It  is  apparent  that  the 
result  is  not  affected  if  the  .90  and  .80  are  interchanged  or 
any  other  interchange  of  the  .80,  .90,  and  .95  is  made. 

The  fact  that  a  change  in  the  order  of  successive  dis- 
counts does  not  change  the  list  price  may  be  very  easily 
shown  by  use  of  algebraic  symbols. 

Finding  Discount  Equivalent  to  Several  Single  Discounts. 

It  is  well  for  the  teacher  to  know  the  short  method  for 
finding  the  single  discount  equivalent  to  two  successive 
discounts. 

Example:  Find  the  single  discount  equivalent  to  the 
successive  discounts  of  20  and  10. 

A  discount  of  20%  leaves  80%  of  the  list  price  to  be 
paid.  A  further  discount  of  10%  leaves  90%  of  the  80% 
of  the  list  price,  or  it  leaves  72%  of  the  list  price  to  be 
paid.  Since  72%  of  the  list  price  is  to  be  paid,  the  two 
discounts  of  20  and  10  must  be  equivalent  to  28%  of  the 
list  price  (100%  of  list  price -72%  of  list  price  =  28% 
of  list  price.) 

This  equivalent  may  readily  be  found  as  follows :  First 
add  the  two  discounts;  then  multiply  them  and  take  .01 
of  their  product;  subtract  this  result  from  their  sum;  the 
remainder  will  be  the  single  discount  equivalent  to  the  two 
successive  discounts.  Applying  the  rule  just  stated  to  the 
discounts,  20  and  10,  we  have  the  following: 

20  +  10  =  30.    20x10  =  200.    .01  of  200  =  2.    30-2  =  28 

A  discount  of  28  is  equivalent  to  discounts  of  20  and  10. 

Find  the  single  discount  equivalent  to  the  successive  dis- 
counts of  15  and  5. 

The  sum  is  20.     The  product  is  75.     .01  of  the  product 


232  HOW  TO  TEACH  ARITHMETIC 

is  .75.     20  -  .75  =  19.25.     Therefore,  a  discount  of  19^  is 
equivalent  to  discounts  of  15  and  5. 

If  required  to  find  the   single   discount   equivalent  to 
three   successive  discounts,   first  find  the  single  discount 
equivalent  to  the  first  two  discounts  and  then  find  the  sin- 
gle discount  equivalent  to  this  result  and  the  third  dis- . 
count. 

For  example:  Find  the  single  discount  equivalent  to 
the  discount  of  20,  10,  and  5. 

Discounts  of  20  and  10  are  equivalent  to  a  single  dis- 
count of  28.  Discounts  of  28  and  5  are  equivalent  to  a 
single  discount  of  31.6.  Therefore,  discounts  of  20,  10, 
and  5  are  equivalent  to  a  single  discount  of  31.6 ;  similarly 

Discounts  of  20,  30  and  10  are  equivalent  to  a  single  dis- 
count of  49.6. 

Why  Discounts  are  Quoted  Separately 

Pupils  sometimes  inquire  why  a  given  firm  does  not 
quote  a  single  discount  instead  of  quoting  two  or  three 
successive  discounts.  Why  shouldn't  a  firm  quote  a  single 
discount  of  31.6  instead  of  three  discounts  of  20,  10,  and 
5?  The  discounts  are  quoted  separately  in  order  that  a 
purchaser  may  take  advantage  of  one  or  two  of  the  dis- 
counts even  though  he  may  be  unable  to  take  advantage 
of  the  three.  The  last  discount  quoted  may  be  ' '  for  cash. ' ' 
The  purchaser  may  be  prepared  to  take  advantage  of  the 
other  discounts  but  he  may  be  unable  to  pay  cash  for  his 
goods.  If  the  three  discounts  were  grouped  as  a  single 
equivalent  discount  a  purchaser  would  not  know  the  de- 
ductions to  be  allowed  for  specific  reasons. 

Illustrative  Problems 

The  list  price  of  some  goods  was  $540.  The  discounts 
were  15,  10,  and  6%.  Find  the  net  price. 


THE  APPLICATIONS  OF  PERCENTAGE  233 

First  Solution 
.85x.90x.94  of  $540  =$388.314  =  net  price. 

Second  Solution 

Discounts  of  15,  10,  and  6  are  equivalent  to  a  single 
discount  of  28.09.  A  discount  of  28.09%  leaves  a  net 
amount  of  71.91%  of  the  list  price  to  be  paid. 

.7191  of  $540  =  $388.314  =  net  price 

A  few  indirect  problems  in  commercial  discount  are 
usually  given  in  text-books.  To  illustrate  : 

What  must  be  the  marked  price  of  goods  to  give  dis- 
counts of  25%,  10%,  and  10%  and  still  realize  $243? 

$243  =  net  price 

Discounts  of  25,  10,  and  10%  are  equivalent  to  a  single 
discount  of  39.25% 

This  leaves  60.75%  of  list  price,  which  =  net  price.. 
Therefore,  60.75%  of  list  price  =  $243. 
Therefore,  list  price  =  $400. 
The  following  is  also  a  solution  for  this  problem  : 

Let  1  =  list  price 

Then  .75  x  .90  x  90  of  1  =  $243 


Therefore  1  =  „,     0A     _r  =  $400 
.75  x.  90  x.  75 

A  merchant  buys  goods  at  discount  of  40%  from  the  list 
price  and  sells  at  a  discount  of  30%  of  list  price  ;  what  per 
cent  does  he  gain  on  the  cost  ? 

The  cost  was  60%  of  the  list  price. 

The  selling  price  was  70%  of  the  list  price. 

The  gain  was  10%  of  the  list  price. 

Since  %  of  gain  is  based  on  the  cost,  we  must  find  what 


234  HOW  TO  TEACH  AEITHMETIC 

%  10  per  cent  of  the  list  price  is  of  60%  of  the  list  price 
(or  the  cost).  It  is  \  of  it,  or  16f%  of  it.  Therefore,  the 
gain  is  16f  %  of  cost. 

At  what  %  above  cost  must  goods  be  marked  in  order 
to  give  a  discount  of  20%  on  the  marked  price  and  still 
make  a  profit  of  30%  on  the  cost? 

Since  a  discount  of  20%  is  given,  the  selling  price  is 
80%  of  the  marked  price. 

Since  a  profit  of  30%  is  made,  the  selling  price  is  130% 
of  the  cost. 

Therefore,  80%  of  marked  price  equals  130%  of  the  cost. 

1%  of  the  marked  price  equals  \3-%  of  the  cost. 

100%  of  the  marked  price  equals  J-^(La%  of  the  cost,  or 
1621%  Of  the  cost. 

Therefore,  the  goods  must  be  marked  62 \%  above  cost. 

It  should  be  noticed  that  no  mathematics  is  involved  in 
the  above  problems  except  the  four  fundamental  operations 
and  the  three  principles  of  percentage  previously  referred 
to.  The  last  problem  solved  involves  the  finding  of  a 
quantity  (the  list  price)  when  a  certain  per  cent  of  it  is 
known. 

Marking  Goods 

This  is  not  a  necessary  part  of  a-  study  of  commercial 
discount  or  of  profit  and  loss,  but  the  pupils  will  be  much 
interested  in  the  subject  and  a  brief  consideration  may  be 
given  to  it. 

Merchants  frequently  indicate  the  cost  price  and  the 
selling  price  on  each  article.  In  order  to  conceal  these  from 
the  customer,  the  merchant  usually  resorts  to  some  symbols 
as  a  private  mark.  Usually  some  word  or  phrase  contain- 
ing ten  different  letters  is  selected  and  used  as  a  "key." 
These  letters  are  used  to  represent  the  nine  digits  and 
zero.  Any  word  or  phrase  of  ten  letters  or  any  ten  ar- 


THE  APPLICATIONS  OF  PERCENTAGE  235 

bitrary  characters  may  be  used  as  a  "key"  or  "cipher." 
An  extra  letter  is  used  to  prevent  the  repetition  of  a  letter, 
and  this  is  called  a  "repeater."  The  repeater  prevents 
giving  any  clew  to  the  private  mark,  as  it  renders  the 
deciphering  more  difficult.  If  the  cost  and  selling  price 
are  both  written  on  the  same  tag  the  selling  price  is  usually 
written  below  and  the  cost  above  a  horizontal  line,  but 
these  positions  are  sometimes  reversed.  Sometimes  the 
cost  price  is  written  in  cipher  known  only  to  the  proprietor. 
Suppose  the  "key"  is  "Pay  us  often"  and  "x"  is  used 
as  a  repeater. 

1234567890        repeater 
payus    often  x 

An  article  which  cost  $4.60  and  is  to  sell  at  $5.40  would 
be  marked 

u.on 
s.un 

An  article  which  cost  $5.56  and  is  to  sell  at  $6.50  would 
be  marked 

s.xo 
o.sn 

The  following  list  of  key  words  and  phrases  may  be 
used  by  the  class.  Select  a  few  of  the  "keys"  and  ask  the 
pupils  to  express  various  costs  and  selling  prices  by  using 
them. 

Importance  Our  Last  Key 

Charleston  Hard  Moneys 

Blacksmith  Buy  for  Cash 

Republican  Cash  Profit 

Buckingham  The  Big  Four 

Gambolines  United  Cars 

Authorizes  Black;  Horse 


236  HOW  TO  TEACH  AEITHMETTC 

Bridgepost  He  Saw  It  Run 

Equinoctial  You  Mark  His 

Frank  Smith  Market  Sign 

Don't  Be  Lazy  Big  Factory 

Now  Be  Quick  No  Suit  Case 

Now  Be  Sharp  Cumberland 

COMMISSION 

The  subject  of  commission  is  studied  in  most  schools  in 
the  seventh  grade ;  in  some  it  is  taught  in  the  sixth  and  in 
others  in  the  eighth  grade. 

Necessity  for  Commission  Business 

The  importance  of  the  commission  business  in  this  coun- 
try is  largely  the  result  of  our  industrial  and  commercial 
development.  Certain  regions  are  recognized  as  centers  for 
particular  products.  Pittsburgh  is  the  center  of  iron  and 
steel  industries ;  Chicago,  Omaha,  and  Kansas  City  are  live 
stock  centers ;  Minneapolis  is  the  center  of  enormous  wheat 
and  flour  industries.  The  farmer  who  ships  his  wheat  to 
Minneapolis  or  the  local  buyer  who  ships  it  for  him  cannot 
always  take  the  shipment  to  the  central  market  and  sell  it 
at  the  best  price.  Economic  conditions  demand  that  there 
shall  be  agents  who  shall  represent  the  buyer  and  seller. 
The  payment  which  this  agent  receives  for  his  services  is 
called  a  commission.  The  commission  is  usually  a  certain 
per  cent  of  the  amount  of  the  sale  or  of  the  purchase; 
sometimes  it  is  a  specified  sum  for  the  performance  of  a 
certain  service. 

The  commission  agent  transacts  business  for  and  in  the 
name  of  another.  The  articles  which  he  buys  or  sells  may 
be  products  of  the  soil,  such  as  wheat,  corn,  or  cotton ;  or 
they  may  be  'live  stock,  farm  or  city  property,  food  stuffs, 
or  numerous  other  things.  The  compensation  paid  to  in- 


THE  APPLICATIONS  OF  PEBCENTAGE  237 

surance  agents,  book  agents,  auctioneers,  buyers  of  farm 
stock  and  tax  collectors  is  usually  called  a  commission.  A 
collector  of  accounts  is  also  said  to  receive  a  commission, 
which  is  based  on  the  amount  collected.  Commission  mer- 
chants are  responsible  to  their  principals  for  the  value  of 
goods  sold  by  them  on  credit.  It  is  not  the  custom  for 
the  commission  merchant  to  charge  a  separate  rate,  for 
assuming  this  risk,  but  the  rate  of  commission  is  large 
enough  to  cover  it.  This  is  one  respect  in  which  a  com- 
mission merchant  differs  from  an  agent. 

Local  Basis  for  Commission 

The  subject  of  commission  may  be  made  interesting  to 
both  city  arid  country  pupils  if  the  introductory  problems 
are  based  on  the  sending  of  farm  products  to  the  city.  In 
almost  every  locality  some  one  is  engaged  in  the  commis- 
sion business  and  these  local  cases  should  be  utilized,  when 
possible,  in  studying  the  subject.  Most  of  the  problems 
should  involve  direct  operations  only  and  all  of  them 
should  be  of  the  types  that  are  in  harmony  with  actual 
business  practice.  The  tax  collector  of  the  community  may 
be  used  as  an  illustration  of  one  whose  compensation  is  a 
commission.  Pupils  should  find  out  the  basis  upon  which 
his  commission  is  computed.  The  technical  terms  of  the 
subject  should  be  understood  by  the  pupil  before  he  at- 
tempts to  solve  the  problems. 

Technical  Terms  of  the  Subject 

Such  fcerms  as  agent,  principal,  consignment,  consignor, 
consignee,  remittance,  net  proceeds,  account  sales  and  ac- 
count purchased  are  technical  words  of  the  subject.  These 
terms  are  defined  in  the  text-books  and  need  no  comment 
here.  The  language  of  commission  usually  causes  the 
pupil  more  difficulty  than  the  mathematics  of  the  subject. 


238  HOW  TO  TEACH  AEITHMETIC 

The  mathematics  of  commission  involves  only  the  funda- 
mental problems  of  percentage. 

Illustrative  Problems 

The  problems  which  follow  illustrate  the  various  types 
that  occur  in  the  subject  of  commission. 

A-  man  places  a  claim  of  $1800  in  the  hands  of  an  attor- 

ney for  collection.    The  attorney  succeeds  in  collecting  only 

60  cents  on  the  dollar.    Find  the  amount  of  the  attorney's 

1  fee  if  he  receives  a  commission  of 


$1800  amount  of  claim 
.60 


_ 

$1080.00  amount  collected 

.025 

5400 
2160 


$27.000  amount  of  fee 

I  paid  an  agent  $55.80  for  buying  wheat  on  a  commis- 
sion of  3%.  Find  the  amount  spent  for  wheat. 

3%  of  the  purchased  price  =  the  commission. 

$55.80  =  the  commission. 

.'.  3%  of  the  purchased  price  =  $55. 80. 

100%  of  the  purchased  price  =  $1860. 

The  proceeds  from  the  sale  of  a  house  were  $3504.  The 
real  estate  agent  received  a  commission  of  4%.  What  was 
the  selling  price  of  the  house  ? 

Since  the  commission  was  4%  of  the  selling  price, 

.'.  the  proceeds  were  96%  of  the  selling  price, 

96%  of  the  selling  price  =  $3504 

100%  of  the  selling  price  =  $3650. 

I  paid  my  agent  5%  for  selling  corn  and  2%  for  invest- 
ing the  net  proceeds  in  wheat.  What  was  the  selling  price 
of  the  corn  if  his  entire  commission  was  $164.50  ? 


THE  APPLICATIONS  OF  PERCENTAGE  239 

100%  of  selling  price  of  corn  =  selling  price  of  corn. 

95%  of  selling  price  of  corn  =  net  proceeds  from  sale. 
(This  includes  the  commission  paid  for  buying  wheat.) 

102%  of  price  paid  for  wheat  =  95%  of  selling  price  of 
corn. 

1%  of  price  paid  for  wheat  =  T9^%.  of  selling  price  of 
corn. 

2%  of  price  paid  for  wheat  =  ff%  of  selling  price  of 
corn.  (This  was  the  second  commission.) 

The  sum  of  the  two  commissions  was  5%  of  selling  price 
of  corn  +  ff%of  selling  price  of  corn,  or  6||%  of  selling 
price  of  corn. 

.'.  6||%  of  selling  price  of  corn  =  $164.50. 

.'.  100%  of  selling  price  of  corn  =  $2397. 

Find  the  amount  of  a  sale  if  an  agent  charged  a  com- 
mission of  3%,  $12.50  for  drayage,  $10.50  for  storage,  and 
$3.50  for  insurance.  The  net  proceeds  of  the  sale  were 
$1389.70. 

The  charges,  exclusive  of  commission,  were  $26.50. 

/.the  amount  of  sale  less  the  commission  was  $1416.20. 

/.  97%  of  the  amount  of  the  sales  =  $1416.20. 

100%  of  the  amount  of  the  sales  =  $1460. 

My  agent  sold  goods  for  $2260;  if  the  commission  was 
$90.40  what  was  the  rate  per  cent  of  commission? 

$90.40  is  4%  of  $2260,  therefore,  the  commission  was  4%. 

SIMPLE  INTEREST 

Pupils  sometimes  fail  to  master  the  subject  of  interest, 
but  the  failure  may  usually  be  traced  to  the  lack  of  an 
accurate  understanding  of  the  terms  used  and  of  an  ac- 
quaintance with  business  procedure  rather  than  to  any 
mathematical  difficulties  involved.  There  is  no  problem  in 
simple  interest  the  solution  of  which  requires  a  degree  of 
mathematical  knowledge  not  in  the  possession  of  the  pupil 


240  HOW  TO  TEACH  AEITHMETIC 

who  is  prepared  to  begin  a  formal  study  of  the  subject. 
Simple  interest  is  an  easy  application  of  percentage  with 
time  as  an  important  factor. 

Numerous  definitions  have  been  suggested  for  the  term 
"interest."  Some  text-books  define  it  as  money  paid  or 
charges  for  the  use  of  money.  The  statement  that  "inter- 
est is  money  rent  "is  probably  as  good  as  any  that  have 
been  proposed. 

The  pupil  should  understand  how  men,  when  lending 
money,  require  a  certain  payment  for  the  use  of  the  money. 
The  teacher  should  make  clear  to  the  pupils  how  a  man 
can  afford  to  borrow  a  given  sum  for  a  year,  be  security 
for  the  amount  borrowed,  and  at  the  end  of  the  year  pay 
back  to  the  lender  not  only  the  amount  originally  bor- 
rowed, but  an  additional  amount,  which  is  called  interest. 
The  sum  upon  which  interest  is  based  is  called  principal, 
to  distinguish  it  from  the  interest,  which  is  of  subordinate 
importance. 

History  of  Interest 

A  brief  historical  survey  of  the  subject  of  "interest"  is 
quite  profitable.  That  the  practice  of  receiving  interest 
should  ever  have  been  regarded  as  immoral  and  as  a  wrong 
to  society,  seems  strange  to  us  because  the  custom  is  so  well 
established;  but  the  propriety  of  such  a  charge  has  been 
frequently  questioned. 

Interest,  or  usury,  as  it  was  formerly  called,  was  charged 
in  the  time  of  the  Babylonians.  From  numerous  refer- 
ences in  the  Bible  we  conclude  that  among  the  early 
Hebrews  it  was  unlawful  to  charge  money  for  the  use  of 
money.  In  later  years  it  was  considered  lawful  to  charge 
a  stranger  usury  (or  interest).  Finally,  it  was  regarded 
as  lawful  to  accept  usury  (or  interest)  from  anyone. 
"Thou  oughtest  therefore  to  have  put  my  money  to  the 


THE  APPLICATIONS  OF  PERCENTAGE  241 

exchangers,  that  at  my  coming  I  should  have  received 
mine  own  with  usury."1  Interest  rates  in  Greece  varied 
from  12%  to  18%;  in  Rome  48%  was  allowed  in  Cicero's 
time,  and  6%  in  the  time  of  Justinian.  The  medieval 
church  was  generally  hostile  to  the  practice  of  charging 
interest.  Italy  was  at  one  time  the  great  financial  center 
of  Europe,  and  the  practice  of  charging  interest  was 
common  there.  In  1552  a  law  was  passed  in  England 
prohibiting  the  charging  of  interest,  and  the  practice  was 
declared  to  be  "contrary  to  the  will  of  God." 

Usury 

The  term  "usury"  means  etymologically  "the  use  of  a 
thing."  The  word  was  originally  applied  to  the  legitimate 
payment  of  money  for  the  use  of  money,  and  was  synon- 
ymous with  the  term  "interest"  as  we  use  the  term  to-day. 

In  most  of  the  countries  where  usury  or  interest  was 
permitted,  laws  were  passed  which  limited  the  rate  that 
might  be  charged.  The  evasion  of  these  laws  by  charging 
excessive  usury  led  to  the  current  use  of  the  term.  To-day 
the  word  usury  means  the  rate  or  amount  of  interest  in 
excess  of  that  permitted  by  law.  In  the  United  States  the 
maximum  rates  of  interest  arc  usually  specified  by  the 
states.  The  maximum  contract  rate  in  New  York,  Penn- 
sylvania, and  New  Jersey  is  6%;  in  Michigan  and 
Illinois  it  is  7%;  in  Ohio  and  Indiana  it  is  8%.  If  a 
citizen  making  a  loan  in  Illinois,  where  the  maximum  rate 
is  7%,  attempts  by  legal  procedure  to  collect  more  than 
7%,  he  is  guilty  of  usury.  In  Ohio  any  rate  higher  than  8% 
would  be  legally  considered  as  usury.  The  pena)ty  for 
charging  usury  differs  in  various  states.  In  some  states 
the  offender  loses  all  of  the  interest;  in  others  he  loses 

i  Matthew  25:27. 


242 


HOW  TO  TEACH  ARITHMETIC 


both  principal  and  interest ;  some  states  provide  no  penalty 
for  usury.  In  several  states  the  usury  laws  have  been 
repealed  and  the  tendency  is  to  allow  an  open  market  for 
capital.  Teachers  may  borrow  a  statute  from  a  justice  of 
the  peace  and  read  the  law  on  interest  and  usury  to  the 
pupils. 

The  following  table  shows  the  interest  rates  and  penalties 
for  usury  prevailing  in  the  states  and  territories : 


Maximum 

States  and 

Legal 

Eate 

Territories. 

Eate. 

Allowed. 

—  Per  cent  — 

Penalty  for  Usury. 

Alabama    

,    8 

8 

Forfeiture  of  all  interest 

Alaska     

,    8 

12 

Arizona     

.   6 

Any 

None 

Arkansas     ...... 

6 

10 

Forfeiture  of  principal  and  interest 

California     

7 

Any 

None 

Colorado     

8 

Any 

None 

Connecticut     

.   6 

6 

None 

Delaware     

6 

6 

Forfeiture  of  double  amount  of 

loan 

Dist.   of  Columbia  6 

10 

Forfeiture  of  all  interest 

Florida     

.   8 

10 

Forfeiture  of  all  interest 

Georgia     

7 

8 

Forfeiture  of  all  interest 

Hawaii     , 

6 

12 

Idaho    

7 

12 

Forfeiture  of  three  times  the  excess 

of  interest  over  12% 

Illinois    , 

5 

7 

Forfeiture   of   all   interest 

Indian  Territory. 

.    6 

8 

Indiana     

.   6 

8 

Forfeiture  of  excess  of  interest 

over 

6% 

Iowa    

6 

8 

Forfeiture  of  all  interest  and  costs 

Kansas    

6 

10 

Forfeiture  of  double  the  excess  of  in- 

terest over  10% 

Kentucky    

,.   6 

6 

Forfeiture  of  excess  of  interest 

Louisiana    

5 

8 

Forfeiture  of  all  interest 

Maine     

6 

Any 

None 

Maryland     

6 

6 

Forfeiture  of  excess  of  interest 

Massachusetts    .  , 

,.   6 

Any 

None 

THE  APPLICATIONS  OF  PERCENTAGE 


243 


Maximum 

States    and 

Legal 

Rate 

Territories. 

Rate. 

Allowed. 

—  Per  cent  —              Penalty  for  Usury. 

Michigan    

5 

7 

Forfeiture  of  all  interest 

Minnesota    

6 

10 

Forfeiture  of  contract 

Mississippi     .... 

.   6 

10 

Forfeiture  of  interest 

Missouri     

.   6 

8 

Forfeiture  of  all  interest 

Montana    

8 

Any 

None 

Nebraska     

7 

10 

Forfeiture  of  all  interest  and  cost 

Nevada    , 

7 

Any 

^None 

New  Hampshire 

.   6 

6 

Forfeiture  of  three  times  the  excess 

of  interest 

New  Jersey    .... 

.   6 

6 

Forfeiture  of  all  interest  and  costs 

New  Mexico     .... 

6 

12 

None 

New  York     , 

6 

6 

Forfeiture  of  principal  and  interest 

North  Carolina     .  , 

,   6 

6 

Forfeiture  of   double  the   amount   of 

interest 

North  Dakota     .  .  . 

7 

12 

Forfeiture  of  all  interest 

Ohio    

6 

8 

Forfeiture  of  excess  over  8% 

Oklahoma     

7 

12 

Oregon     

6 

10 

Forfeiture  of  interest,  principal  and 

costs 

Pennsylvania     .  .  . 

6 

6 

Forfeiture  of  excess  of  interest 

Philippine  Islands 

6 

Any 

Porto  Rico    

12 

12 

Rhode  Island     .... 

6 

Any 

None 

South  Carolina     .  . 

7 

8 

Forfeiture  of  all  interest 

South  Dakota    .  .  . 

7 

12 

Forfeiture  of  all  interest 

Tennessee    

6 

6 

Forfeiture  of  excess  of  interest 

Texas    

6 

10 

Forfeiture  of  all  interest 

Utah    

8 

Any 

None 

Vermont     

6 

6 

Forfeiture  of  excess  of  interest 

6 

6 

Forfeiture  of  excess  of  interest 

over  8% 

Washington    

10 

12 

Forfeiture  of  double  illegal  interest 

West  Virginia    .  .  . 

6 

6 

Forfeiture  of  excess  of  interest 

Wisconsin     

6 

1C 

Forfeiture  of  all  interest 

Wyoming     

8 

12 

None 

244  HOW  TO  TEACH  ARITHMETIC 

Why  Maximum  Interest  Laws  Were  Enacted 

Teachers  should  consider  the  conditions  which  gave  rise 
to  the  enactment  of  maximum  interest  laws  by  many  of 
the  states.  The  laws  allow  an  open  market  to  capital 
which  is  invested  in  houses  or  farms  or  merchandise.  An 
owner  is  permitted  to  secure  for  his  property  whatever 
rent  the  laws  of  supply  and  demand  will  permit.  If  a 
man  has  his  capital  in  the  form  of  money  and  wishes  to 
rent  it,  that  is,  to  charge  interest  for  it,  why  should  the 
state  specify  the  maximum  rent  which  he  may  legally 
receive?  Is  the  borrower  any  more  at  the  mercy  of  the 
unprincipled  money  lender  than  the  renter  is  at  the  mercy 
of  those  who  seek  to  charge  exorbitant  rents?  Is  he  not 
liable  to  be  a  victim  of  extortion  in  either  case?  What 
reason  is  there  for  regulating  interest  rates  that  will  not 
apply  to  the  regulating  of  rents  for  houses  or  farms  ?  The 
answer  to  this  question  will  doubtless  be  suggested  to  the 
thoughtful  teacher  when  he  considers  those  who  needed 
protection  from  exorbitant  interest  rates  when  the  laws 
were  enacted  and  those  who  are  the  great  borrowers  of 
capital  to-day. 

Many  states  not  only  specify  the  maximum  rate  of 
interest  that  may  be  legally  collected,  but  the  " legal  rate" 
is  also  determined  by  legislative  enactment.  The  "  legal 
rate"  means  that  rate  of  interest  which  may  be  legally 
collected  when  the  words  "with  interest"  are  included  in  a 
note  but  no  rate  of  interest  is  specified.  The  legal  rate  is 
also  the  rate  which  may  be  collected  on  a  note  without 
interest  which  is  not  paid  when  due.  Such  a  note  draws 
the  legal  rate  of  interest  from  the  date  of  maturity  until 
it  is  paid.  It  will  be  seen  from  the  preceding  table  that 
in  some  states  the  legal  rate  is  the  same  as  the  maximum 
rate ;  in  others  it  is  less. 


THE  APPLICATIONS  OF  PERCENTAGE  245 

How  Interest  Rates  Are  Determined 

The  pupils  should  consider  the  various  factors  which 
determine  interest  rates.  The  relation  of  interest  rates  to 
the  law  of  supply  and  demand  for  money  should  be  pointed 
out.  Other  things  being  equal,  interest  rates  are  low  when 
the  amount  of  money  to  be  lent  exceeds  the  demands  of 
those  who  wish  to  borrow ;  rates  are  high  when  the  demands 
of  the  borrowers  exceed  the  amount  of  money  to  be  lent. 
Especial  attention  should  be  directed  to  the  security  of  the 
loan  as  a  factor  in  determining  interest  rates.  The  United 
States  Government  can  borrow  large  sums  of  money  at  low 
rates  of  interest.  An  unstable  government  must  pay  high 
interest  rates.  It  is  the  duty  of  the  teacher  to  impress 
upon  the  pupils  the  fact  that  very  high  rates  of  interest 
are  often  synonymous  with  poor  security.  The  first  thing 
to  be  investigated  in  making  a  loan  is  not  the  rate  of 
interest  that  is  charged,  but  the  security  of  the  loan.  It 
would  not  be  correct  to  say  that  high  rates  of  interest  are 
always  directly  associated  with  poor  security,  for  where 
profits  upon  capital  are  large,  the  rates  of  interest  are 
high  as  a  result  of  the  law  of  supply  and  demand.  How- 
ever, the  teacher  should  caution  the  pupil  to  investigate 
with  more  than  usual  care  the  security  of  any  loan  when 
very  high  returns  are  promised.  In  this  connection  some 
consideration  should  be  given  to  the  pernicious  practice  of 
the  "loan  sharks."  These  people  often  secure  the  pay- 
ment of  exorbitant  rates  of  interest  and  the  unfortunate 
man  or  woman  who  becomes  financially  obligated  to  a 
"loan  shark"  finds  it  next  to  impossible  to  discharge  his 
so-called  obligation  and  free  himself  from  the  clutches  of 
his  oppressor.  In  recent  years  the  courts  are  becoming 
more  active  in  condemning  the  impossible  contracts  which 
these  lenders  induce  their  victims  to  sign. 


246  HOW  TO  TEACH  AKITHMETIC 

A  third  factor  that  determines  interest  rates  is  the  time 
for  which  the  loan  is  made.  The  rate  is  usually  lower 
upon  a  loan  for  a  long  period  than  for  a  short  period.  The 
amount  of  the  loan  is  also  a  factor  in  determining  interest 
rates.  The  rate  for  a  small  loan  is  often  higher  than  for  a 
large  one. 

Days  of  Grace 

Owing  to  the  rapid  growth  of  banking  facilities,  short 
term  notes  have  become  so  general  that  the  teaching  of 
interest  is  gradually  becoming  simplified  in  the  schools.  It 
is  now  customary  to  pay  interest  every  30,  60,  or  90  days, 
or  else  every  year.  Days  of  grace  have  been  abolished  in 
many  of  the  states  because  the  development  of  banking 
facilities  has  rendered  them  unnecessary.  The  teacher 
should  ascertain  whether  days  of  grace  are  still  permitted 
in  his  state ;  and  if  not  permitted,  they  should  be  omitted 
from  all  problems,  irrespective  of  the  statement  in  the 
text-book. 

Kinds  of  Interest 

Since  the  year  contains  approximately  365J  days,  there 
is,  strictly  speaking,  no  such  thing  as  exact  interest.  How- 
ever, the  name  exact  interest  is  applied  to  simple  interest 
computed  upon  the  exact  number  of  days  of  each  month 
and  considering  the  year  as  consisting  of  365  days. 

In  computing  bankers'  interest  the  exact  number  of  days 
of  each  month  is  reckoned,  but  the  360-day  year  is  used. 
In  common  interest  360  days  are  considered  a  year,  30 
days  a  month,  and  a  month  TV  of  a  year. 

Exact  interest  is  used  by  the  United  States  Government, 
by  some  large  banks,  and  in  finding  the  interest  on  foreign 
money.  Bankers'  interest  is  used  by  most  banks  and  by 
business  men  in  finding  the  interest  on  short  time  notes. 

Common  interest  is  used  in  finding  the  interest  on  notes 


THE  APPLICATIONS  OF  PERCENTAGE  247 

and  debts  that  bear  interest  for  longer  periods  of  time- 
usually  when  the  time  is  a  year  or  more. 

Exact  interest  for  any  number  of  days  may  be  found  by 
taking  ^f  of  the  amount  of  the  common  interest.  (£f$  is 
?l  °f  Iff)-  Common  interest  may  be  found  from  exact 
interest  by  increasing  the  exact  interest  by  -fa  of  itself. 
(Ill  is  I!  of  m). 

If  we  compare  the  exact,  bankers',  and  common  interest 
on  a  given  principal  at  a  given  rate  for  an  interval  of 
several  months,  we  will  find  that  the  bankers'  interest  is 
slightly  higher  and  the  exact  interest  is  slightly  lower  than 
the  common  interest. 

Methods  of  Computing  Interest 

There  are  numerous  methods  for  computing  simple 
interest.  One  or  more  of  the  following  methods  are  usually 
considered  in  text-books  on  arithmetic:  Six  Per  Cent. 
Aliquot  Part,  Twelve  Per  Cent,  Formula,  Cancellation, 
One  Per  Cent,  Thirty-six  Per  Cent,  Dollar,  Month,  One 
Day,  Bankers',  Sixty  Day.  Pupils  should  be  taught  only 
one  method  besides  the  general  method  for  computing 
interest.  The  attempt  to  develop  a  mastery  of  several 
methods  usually  results  in  confusion.  It  is  much  better 
to  secure  a  mastery  of  one  good  method  than  to  have  a 
partial  mastery  of  several.  There  is  some  difference  of 
opinion  as  to  the  best  single  method  for  computing  interest, 
but  the  tendency  in  recent  years  seems  to  be  towards  the 
six  per  cent,  the  aliquot  part,  and  the  formula  methods, 
although  a  number  of  the  others  have  commendable 
features. 

The  Six  Per  Cent  Method 

The  six  per  cent  method  is  one  of  the  shortest  for  finding 
the  interest  on  $1  for  a  given  number  of  days.  Since  it 


248  HOW  TO  TEACH  AEITHMETIC 

is  based  on  a  year  of  360  days,  it  is  somewhat  inexact. 
However,  in  many  financial  transactions  the  difference 
between  the  interest  computed  on  the  basis  of  a  year  of 
360  days  and  that  computed  on  the  basis  of  365  days  is 
negligible. 

The  six  per  cent  method  is  based  on  the  following  facts : 

Interest  on  $1  at  6%  for  1  year  (360  days)  =  $.06. 

Interest  on  $1  at  6%  for  2  mo.  (i  of  1  yr.)  =  $.01. 

Interest  on  $1  at  6%  for  1  mo.  (^  of  2  mo.)  =  $.005. 

Interest  on  $1  at  6%  for  6  da.  (i  of  1  mo.)  =  $.001. 

Interest  on  $1  at  6%  for  1  da.  (£  of  6  da.)  =  $.000£. 

It  is  readily  seen  that  the  interest  on  $1  at  6%  for  1 
month  is  ^  cent.  The  interest  on  $1  at  6%  for  any  number 
of  months  is  half  as  many  cents  as  there  are  months.  Thus 
the  interest  for  8  mo.  is  $.04;  for  7  mo.  it  is  $.035;  for 
11  mo.  it  is  $.055. 

It  is  evident  also  that  since  the  interest  on  $1  at  6%  for 
1  day  is  4  of  a  mill,  the  interest  on  $1  at  6%  for  any  num- 
ber of  days  will  be  £  as  many  mills  as  there  are  days. 
Thus  the  interest  on  $1  at  6%  for  18  da^  is  $.003;  for 
21  da.  it  is  $.0035. 

The  above  facts  enable  us  to  determine  readily  the 
interest  on  $1  at  6%  for  any  number  of  months  or  days. 
Required  to  find  the  simple  interest  on  $480  for  7  mo. 
18  da.  at  6%. 

The  interest  on  $1  at  6%  for  7  mo.  18  da.  is  $.038. 

.*.  the  interest  on  $480  at  6%  for  7  mo.  18  da.  is  480  x 
$.038  =  $18.24. 

If  the  rate  above  were  4%  we  would  take  £  of  the  interest 
just  found.  If  the  rate  were  3^%  we  would  'divide  the 
above  interest  by  6  (which  would  give  the  interest  at  1%) 
and  then  multiply  this  result  by  3^. 


THE  APPLICATIONS  OF  PERCENTAGE  249 

The  Aliquot  Part  Method 

In  computing  interest  by  the  Aliquot  Part  Method  the 
time  is  separated  into  two  or  more  parts  so  that  each  part 
less  than  a  year  is  a  unit  fraction  of  some  preceding  part. 
One  illustration  will  make  the  method  clear. 

Required  to  find  the  simple  interest  on  $480  for  1  yr. 
7  mo.  18  da.  at  5%. 

$480 
.05 


$24.00  Int.  for  1  yr. 

6  mo.  =  \  yr.  $12.       Int.  for  6  mo. 

1  mo.  =  £  of  6  mo.     $  2.       Int.  for  1  mo. 

15  da.  =  -J  mo.  $  1-       Int.  for  15  da. 

3  da  =  TV  mo.  $  0.20  Int.  for  3  da. 

$39.20  Int.  for  1  yr.  7  mo.  18  da. 

The  Aliquot  Part  Method  is  not  applicable  when  accu- 
rate interest  is  desired. 


The  Formula  Method 

The  chief  objection  urged  against  the  formula  method  is 
that  the  principles  underlying  the  calculation  of  interest 
are  so  elementary  that  there  is  no  occasion  to  substitute 
mechanical  rules  for  the  application  of  a  few  simple  prin- 
ciples. If  the  pupil  knows  the  general  formula  I  =  p  **  — 

and  .can  solve  a  simple  equation  he  can  use  the  formula  to 
advantage.  If  the  habit  of  employing  equations  in  the 
solution  of  problems  has  been  formed,  such  formulas  as  the 
above  may  be  used  to  advantage. 


250  HOW  TO  TEACH  ARITHMETIC 

Problems  in  Simple  Interest 

Many  of  the  problems  in  simple  interest  should  be  solved 
orally  and  the  pupil  should  be  encouraged  to  use  the  short- 
est method  to  secure  the  desired  result.  In  general,  it  is 
better  to  postpone  the  consideration  of  short  methods  until 
the  pupils  are  reasonably  familiar  with  the  general  methods. 

The  six  per  cent  method  may  be  considerably  abridged, 
^nd  there  is  no  reason  why  the  short  form  should  not  be 
taught  to  the  pupil  after  he  is  familiar  with  the  more 
expanded  form. 

If  required  to  find  the  simple  interest  on  $420  for  5  mo. 
18  da.  at  6%,  we  may  proceed  as  follows: 

SOLUTION  : 

168 x $420 
6x1,000  - 

The  168  is  derived  from  the  reduction  of  5  mo.  18  da.  to 
days.  We  divide  by  6  because  the  principle  requires  us 
to  take  ^  of  the  number  of  days.  We  divide  by  1,000 
because,  having  taken  ^  of  the  number  of  days,  we  must 
use  the  result  as  mills,  and  mills  stands  in  thousandths 
place. 

What  is  the  simple  interest  on  $480  at  1%  for  5  mo. 
12  da.  ? 

162  x  7  x  $480 


6  x  6  x  1000 


-  =  $15.12 


We  divide  by  the  first  six  because  we  must  take  ^  of  the 
162  days;  by  the  second  6  in  order  to  find  the  interest 
at  1% ;  we  multiply  this  result  by  7  in  order  to  obtain  the 
interest  at  7%. 

One  more  problem  will  illustrate  sufficiently  this  abridged 
form.  Find  the  simple  interest  on  $840.24  for  3  mo.  24  da. 
at  5%. 


THE  APPLICATIONS  OF  PERCENTAGE  251 

3  mo.  24  da.  =  114  da. 

114  x  5  x  $840.24 


6  x  6  x  1000 


-=$13.303 


One  advantage  of  this  method  is  the  opportunity  afforded 
for  cancellation. 


Computing  Interval  between  Dates 

In  computing  the  time  between  two  given  dates, 
the  teacher  should  require  the  pupils  to  use  the  method 
prevalent  in  the  community.  In  some  localities  the 
time  is  computed  in  days  when  the  interval  is  less  than 
a  year,  and  in  years,  months,  and  days  when  the  time  is 
greater  than  a  year.  In  other  localities  the  time  for  periods 
greater  than  a  year  is  found  in  years  and  days.  The  time 
from  April  14,  1910,  to  November  8,  1912,  may  be  com- 
puted by  any  of  the  following  methods : 

(a)     1912     11      8 

1910      4    14 

2      6    24 

The  result  is  2  yr.  6  mo.  24  da. 

(b)  The  time  from  April  14,  1910,  to  April  14,  1912,  is 
2  years.    From  April  14  to  October  14  is  6  months.    From 
October  14  to  November  8  is  25  days.     The  result  is  2  yr. 
6  mo.  25  da. 

(c)  From  April  14,  1910,  to  April  14,  1912,  is  2  years. 
From  April  14  to  November  8  (counting  the  exact  number 
of  days  in  each  month)  is  208  da.,  or  6  mo.  28  da.     The 
entire  time  is  therefore  2  yr.  6  mo.  28  da.    There  are  sec- 
tions of  the  United   States  in  which  each  of  the  three 
methods  is  recognized  as  correct.    In  some  states  both  the 


252  HOW  T0  TEACH  AEITHMETIG 

day  a  note  is  given  and  the  day  it  matures  are  considered 
in  computing  the  time. 

Some  texts  contain  a  table  showing  the  number  of  days 
from  any  day  of  one  month  to  the  same  day  of  any  other 
month  within  a  year.  Such  a  table  is  found  on  page  275. 

Interest  Table 

If  time  permits  teachers  may  interest  pupils  in  construct- 
ing a  portion  of  an  interest  table,  such  as  is  frequently 
employed  by  bankers.  A  portion  of  such  a  table  is  given 
below. 

2  months  4% 

Total 

days.       $1000  $2000  $3000  $4000  $5000  $6000  $7000  $8000  $9000 

60 6.66  13.33  19.99  26.65  33.33  39.99  46.66  52.22  59.99 

61 6.77  13.55  20.33  27.10  33.89  40.66  47.43  54.21  60.99 

62 6.88  13.77  20.66  27.55  34.44  41.32  48.21  55.10  61.99 

The  table  shows  that  the  interest  on  $3000  for  60  days 
at  4%  is  $19.99.  The  interest  on  $300  for  60  days  is 
$1.999  and  on  $30  it  is  $.1999.  The  interest  on  $7000  for 
62  days  at  4%  is  $48.21.  By  moving  the  decimal  point  the 
interest  can  easily  be  found  on  $700,  $70,  $7,  or  $0.70 
at  4%. 

By  using  the  table  we  may  find  the  interest  on  $5486  for 
61  days  at  4%  as  follows: 

The  interest  on  $5000  for  61  days  at  4%  is  $33.89. 

The  interest  on  $400  for  61  days  at  4%  is  $2.71. 

The  interest  on  $80  for  61  days  at  4%  is  $0.5421. 

The  interest  on  $6  for  61  days  at  4%  is  $0.0406. 

Therefore,  the  interest  on  $5486  for  61  days  at  4%  is 
$37.1827,  or  $37.18. 

Similar  tables  may  be  constructed  for  any  number  of 
days  and  at  any  desired  per  cent.  Such  tables  are  much 


THE  APPLICATIONS  OF  PERCENTAGE  253 

used  by  banks,  insurance  offices  and  trust  companies.  The 
tables  are  arranged  in  convenient  form  and  they  greatly 
lessen  the  labor  of  computing  interest.  Several  different 
arrangements  of  interest  tables  are  published. 

Indirect  Problems 

The  "indirect"  problems  in  interest  have  a  very  limited 
application  and  many  of  them  should  be  omitted.  The 
important  thing  is  to  find  the  interest.  There  is  a  grow- 
ing tendency  to  devote  attention  to  the  real  problems  only 
in  studying  interest  and  to  use  the  time  thus  saved  in 
writing  promissory  notes  and  in  familiarizing  the  pupil 
with  other  business  procedures.  Genuine  mercantile  trans- 
actions supply  problems  that  are  sufficiently  complex  for 
the  average  pupil.  It  is  certain  that  the  indirect  problems 
in  interest,  if  taught,  should  not  be  taught  by  rules.  It  is 
practically  useless  for  the  pupils  to  work  the  problems  in 
a  mechanical  way.  The  problems,  if  considered  at  all, 
should  be  taught  so  as  to  afford  a  training  that  is  worth 
while.  A  few  of  the  indirect  problems  will  be  considered. 

1.  The  interest  on  $420  for  2  years,  3  months  is  $47.25, 
what  is  the  rate? 

a) 

2  yr.  3  mo.  =  f  yr. 
The  interest  for  £  yr.  =  $47.25 
The  interest  for  1  yr.  =  f  of  $47.25  =  $21 
/.  r%  of  $420  -$21 


The  above  problem  may  be  solved  as  follows  : 
b)   The  interest  on  $420  for  2  years,  3  months  at  1%  = 
$9.45;  therefore,  to  produce  $47.25  in  the  same  time  the 


254  HOW  TO  TEACH  AEITHMETIC 

rate  must  be  as  many  times  1%  as  $47.25  is  times  $9.45. 
$47.25  is  5  times  $9.45 ;  hence  the  required  rate  must  be 
5% ;  or  it  may  be  solved  as  follows: 
c) 

Let  r%=the  rate 
r%x2Jx  $420  =  $47.25 

„     $47.25     _.. 
••r%  =  $9!<r  =  5% 

2.  How  long  will  it  take  the  interest  on  $480  at  6%  to 
equal  $48? 

a) 

The  interest  for  1  yr.  =  .06  of  $480 

The  interest  for  t  yr.  =  tx.06  of  $480 

. \tx.06  of 

.".  the  time  is  If  yr.,  or  1  yr.  8  mo. 
The  above  problem  may  be  solved  as  follows: 
b)   The  interest  on  $480  for  1  year  at  6%  is  $28.80.    To 
produce  $48  interest  at  the  same  rate,  the  time  must  be 
as  many  times  one  year  as  $48  is  times  $28.80.    $48  is  If 
times  $28.80;  hence,  the  time  is  If  year,  or  1  year  and 
8  months. 

3.  On  what  sum  of  money  will  the  interest  for  1  year, 
4  months,  at  6%,  equal  $57.60? 

a)  Let  $p  =  the  principal. 
The  interest  on  $p  for  f  yr.  at  6%  is  $57.60 
Therefore  the  interest  on  $p  for  1  yr.  at  6%  is  f  of 
$57.60,  or  $43.20 

therefore  $px.06  =  $  43.20 


or  we  may  say : 


THE  APPLICATIONS  OF  PERCENTAGE  255 

(b) 

£  x.  06  x$p  =  $57.60 
$57.60     $57.60 


or  as  follows: 

c)  A  principal  of  $1  will  produce  $.08  interest  in  1  yr. 
4  mo.  at  6%.  To  produce  $57.60  interest  the  principal 
must  be  as  many  times  $1  as  $57.60  is  times  $.08.  $57.60 
is  720  times  $.08,  hence  the  required  principal  is  $720. 

4.  Find   the  principal  which  will   amount   to   $562.68 
in  8  mo.  12  da.  at  6%. 
a) 

p  =  principal 

p  +  .042p  =  $562.68  (.042  is  found  by  getting 

/.  1.042p  =  $562.68  the  interest  on  $1  at  6%  for  8 

$562.68        ,.A      mo.  12  da.) 


or  as  follows: 

4.  (b)  A  principal  of  $1  will  amount  to  $1.042  in  8 
mo.  12  da.  at  6%.     To  amount  to  $562.68  the  principal 
must  be   as  many  times  $1   as  $562.68   is  times  $1.042. 
$562.68  is  540  times  $1.042;  hence,  the  required  princi- 
pal is  $540. 

5.  In  what  time  will  any  principal  double  itself  at  5% 
simple  interest? 

To  double  itself  a  principal  must  gain  100%  of  itself. 
Since  in  1  yr.  the  principal  gains  5%  of  itself,  it  will  re- 
quire 20  years  to  gain  100%  of  itself. 

6.  At  what  rate  will  any  principal  double  itself  in  25 
years  ? 

Since  a  principal  gains  100%  of  itself  in  25  yr.  ;  .*.  it 
gains  ^  of  100%  of  itself  in  1  yr.  The  yearly  gain  is 
4%,  which  is,  therefore,  the  rate. 


256  HOW  TO  TEACH  AEITHMETIC 

ANNUAL  INTEREST 

Annual  interest  is  interest  payable  annually  or  at  any 
other  regular  interval,  as  semi-annually,  or  quarterly.  If 
annual  interest  is  not  paid  when  due  it  draws  simple  inter- 
est from  the  time  it  becomes  due  until  it  is  paid.  Annual 
interest  may  be  defined  as  simple  interest  on  the  principal, 
increased  by  the  simple  interest  on  each  interval's  interest 
from  the  close  of  the  interval  to  the  time  of  settlement. 
The  interval  may  be  one  year  or  one-half  year,  or  one- 
fourth  of  a  year,  or  any  other  specified  period.  If  the 
interest  is  payable  semi-annually  or  quarterly  it  is  com- 
puted in  the  same  manner  as  when  it  is  payable  annually. 

Since  the  intervals  involved  in  this  type  of  interest  are 
sometimes  fractional  parts  of  a  year  the  term  "periodic 
interest"  is  frequently  used  instead  of  annual  interest. 

In  many  states  unpaid  interest  does  not  draw  interest 
until  settlement,  but  annual  interest  is  legalized  in  Michi- 
gan, Ohio,  Wisconsin,  Vermont,  New  Hampshire,  and 
Iowa.  In  Pennsylvania,  Georgia,  Illinois,  and  Indiana  it 
is  legal  by  special  contract  only.  It  is  the  custom  in  some 
parts  of  the  country  to  draw  up  a  note  for  the  principal 
without  interest  for  the  specified  time  and  interest  notes 
which  mature  at  the  time  each  interest  payment  is  payable. 

These  interest  notes  provide  for  the  payment  of  simple 
interest  on  all  unpaid  interest. 

If  annual  interest  is  not  permitted  in  a  given  state,  it 
should  not  be  taught  in  the  schools.  If  it  is  permitted  in 
the  state,  the  teacher  should  ascertain  the  rate  at  which 
it  is  legal  and  should  use  that  rate  in  the  problems. 

Illustrative  Problems 

The  solution  of  one  problem  will  suffice  to  illustrate  this 
type  of  interest. 


THE  APPLICATIONS  OF  PERCENTAGE  257 

Indianapolis,  Ind.,  June  18,  1913. 
$600. 

Four  years  after  date  I  promise  to  pay  to  the  order  of 
F.  S.  Lint  Six  Hundred  Dollars,  value  received,  with  in- 
terest at  5%,  payable  annually. 

THOS.  W.  BRIGGS. 

If  no  interest  is  paid  until  this  note  matures,  how  much 
is  then  due? 

Solution.  The  interest  on  $600  for  4  years  at  5%  is 
$120. 

The  interest  for  the  first  year  is  $30  and  this  will  draw 
interest  at  5%  for  3  years.  (Since  the  note  does  not  ma- 
ture for  3  years  after  the  first  interest  is  due.)  The  inter- 
est for  the  second  year  will  draw  interest  for  2  years  and 
the  interest  of  the  third  year  for  1  year.  Thirty  dollars 
will,  therefore,  draw  interest  at  5%  for  3  years  plus  2 
years  plus  1  year,  or  6  years.  The  interest  on  $30  for 
6  years  at  5%  is  $9.  The  amount  due  when  the  note 
matures  is,  therefore,  $600  plus  $120  plus  $9,  or  $729. 

At  5%  simple  interest  $100  would  amount  to  $160  in 
ten  years;  at  annual  or  periodic  interest  it  would  amount 
to  $176.20. 

COMPOUND  INTEREST 

Compound  interest  is  the  interest  that  accrues  by  mak- 
ing the  interest  due  at  the  close  of  any  interval,  for  which 
the  interest  is  made  payable,  a  part  of  the  interest  bearing 
debt  for  the  next  succeeding  interval.  In  other  words  the 
entire  amount  due  at  the  end  of  any  interval  becomes  the 
principal  for  the  next  interval.  Interest  may  be  com- 
pounded annually,  semi-annually  or  quarterly,  according 
to  agreement.  In  most  of  the  states  the  collection  of  com- 
pound interest  cannot  be  enforced  by  law. 


258  HOW  TO  TEACH  AEITHMETIC 

Comparison  of  Simple,  Annual  and  Compound  Interest 

The  essential  difference  between  annual  and  compound 
interest  is  that  in  annual  interest  the  interest  for  a  given 
interval  draws  simple  interest  until  the  day  of  settlement, 
while  in  compound  interest  it  draws  compound  interest 
until  the  day  of  settlement.  The  amount  of  $100  at  6% 
annual  interest  for  ten  years  would  be  $176.20.  At  com- 
pound interest  it  would  be  $179.08,  while  at  simple  inter- 
est the  amount  would  be  $160.  Money  loaned  at  compound 
interest  increases  with  great  rapidity.  A  given  principal 
will  more  than  double  itself  in  12  years  at  6%  compound 
interest. 

The   following   solution   will   illustrate   the   method   of 
computing  compound  interest.     Find  the  compound  inter- 
est on  $500  for  3  years,  3  months,  15  days  at  4%,  interest 
compounded  annually. 
SOLUTION  :     $500  =  principal  for  first  year 

.04 

$  20  interest  for  first  year 
$500 
$520  amt.  for  1st  yr.  =  principal  for  2d  yr. 

.04 

$  20.80  interest  for  second  year 
$520 

$540.80  amount  for  2d  yr.  =  principal  for  3d  yr. 
.04 


$  21.6320  interest  for  third  year 
$540.80 

$562.432    amount  for  3d  yr.  =  principal  for  4th  yr. 
The  interest  on  $562.432  for  3  mo.  15  da.  at  4%  is  $6.56. 
$562.43  +  $6.56  =  $568.99  =  amt.  for  3  yr.  3  mo.  15  da. 
$500 
$  68.99  comp.  int.  for  3  yr.  3  mo.  15  da. 


THE  APPLICATIONS  OF  PERCENTAGE  259 

When  compound  interest  is  payable  semi-annually  or 
quarterly,  we  find  the  amount  of  the  given  principal  for 
the  first  interval,  and  make  it  the  principal  for  the  second 
interval,  etc.  When  the  principal  contains  years,  months 
and  days,  as  in  the  problem  above,  we  find  the  amount  for 
the  nearest  exact  interval  and  upon  this  amount  compute 
the  interest  for  the  remaining  months  and  days.  This  is 
then  added  to  the  last  amount  before  subtracting  the  origi- 
nal principal. 

The  chief  use  of  compound  interest  is '  among  savings 
banks,  building  and  loan  associations,  private  banking 
houses  and  insurance  companies.  In  practice  a  compound 
interest  table  is  generally  used. 

A  section  of  a  compound  interest  table  will  illustrate 
the  use.  A  table  may  be  computed  to  any  desired  degree 
of  accuracy. 

The  following  table  shows  the  compound  amount  of  $1  for 
intervals  of  1  to  10  years : 


Years. 
1.  .  . 

,  .  .  1 

2% 
.0200 

2 
1 

:V2% 
.0250 

e 
1 

\% 

.0300 

3 
1 

%% 
.0350 

4% 
1.0400 

41/2% 
1.0450 

5% 
1  0500 

9  

...  1 

0404 

1 

.0506 

1 

.0609 

1 

.0712 

1.0816 

1.0920 

1  1025 

3  

...  1 

.0612 

1 

.0768 

1 

.0927 

1 

.1087 

1.1248 

1.1411 

1.1576 

4  

...  1 

.0824 

1 

.1038 

1 

.1255 

1 

.1475 

1.1698 

1.1925 

1.2155 

5  

.  .  .  1 

1040 

1 

.1314 

1 

.1592 

1 

.1876 

1.2166 

1.2461 

1.2762 

6  

...  1 

.1261 

1 

.1596 

1 

.1940 

1 

.2292 

1.2653 

1.3022 

1.3400 

/  

...  1 

.1486 

1 

.1886 

1 

.2298 

1 

.2722 

1.3159 

1.3608 

1.4007 

8  

...  1. 

,1716 

1 

.2184 

1 

.2667 

1 

.3168 

1.3685 

1.4221 

1.4774 

9  

..  .  1, 

,1950 

1 

.2488 

] 

.3047 

1 

.3628 

1.4233 

1.4860 

1.5513 

10... 

.  1. 

,2189 

1 

.2800 

1 

.3439 

1 

.4105 

1.4802 

1.5529 

1.6288 

The  amount  on  $1  for  4  years  at  4%  is  $1.1698. 

The  amount  of  $300  at  4%  for  4  years  interest  com- 
pounded annually  is  1.1698  x  $300;  the  amount  of  $548.60 
for  2  years  at  3%  is  1.0609  x  $548.60. 

The  compound  amount  of  any  given  sum  at  a  given  rate 


260  HOW  TO  TEACH  ARITHMETIC 

payable  semi-annually,  equals  the  compound  amount  of 
the  same  sum  for  twice  the  time  at  one-half  the  rate  pay- 
able annually.  The  compound  amount  of  any  sum  at  a 
given  rate  payable  quarterly  equals  the  compound  amount 
of  the  same  sum  for  four  times  the  time,  at  one-fourth 
the  rate,  payable  annually.  For  example,  3  yr.  6  mo.  at 
4%  semi-annually  will  yield  the  same  compound  amount 
as  if  the  rate  is  2%  for  7  years  and  the  interest  is  com- 
pounded annually.  The  compound  amount  of  a  given  sum 
for  3  years  at  8%  compounded  quarterly  is  the  same  as 
for  12  years  at  2%  compounded  annually. 

Applications 

The  subject  of  compound  interest  affords  the  teacher  an 
excellent  opportunity  to  direct  attention  to  savings  banks 
and  the  importance  of  these  institutions.  If  the  state  laws 
governing  savings  banks  are  rigid  enough  to  make  them 
desirable  institutions  the  teacher  should  direct  attention 
to  the  fact  that  such  institutions  receive  small  as  well  as 
large  deposits  and  pay  interest  upon  all  deposits  left  for 
a  certain  length  of  time.  An  appeal  to  economy  and  thrift 
may  be  made  by  calling  attention  to  the  rapidity  with 
which  even  small  deposits  increase  if  they  are  regularly 
made  and  left  in  the  bank  at  interest  for  a  few  years. 

Ask  the  pupils  to  compute  how  much  a  boy  would  have 
in  a  savings  bank  at  the  end  of  5  or  10  years  if  he  deposited 
$1  each  week  and  made  no  withdrawals,  the  bank  paying 
3%  interest,  compounded  quarterly.  Most  pupils  will  be 
much  surprised  at  the  rapidity  with  which  even  small  sav- 
ings accumulate. 

The  following  illustration  is  taken  from  "A  Scrap  Book 
of  Elementary  Mathematics,"  by  William  F.  "White  (p. 
47).  It  shows  the  enormous  results  obtained  when  com- 


THE  APPLICATIONS  OF  PERCENTAGE  261 

pound  interest  is  computed  for  long  periods.     The  illus- 
tration will  impress  this  fact  upon  the  minds  of  pupils. 

"At  3%  one  dollar  put  at  interest  at  the  beginning  of 
the  Christian  era  to  be  compounded  annually  would  have 
amounted  in  1906  to  ($1.03)1906,  which,  in  round  num- 
bers, is  $3,000,000,000,000,000,000,000,000.  The  amount  of 
$1  for  the  same  time  and  rate,  but  at  simple  interest,  would 
be  only  $58.18." 

INSURANCE 

Insurance  is  such  an  important  factor  in  modern  life 
that  the  broader  aspects  of  the  subject  should  be  taught 
in  the  schools.  No  attempt  should  be  made  to  explain  the 
technicalities  involved  in  the  various  kinds  of  policies. 
The  informational  value  of  the  subject  should  be  emphasized 
rather  than  its  mathematical  content.  Only  the  common 
types  of  property  and  personal  insurance  should  be 
considered. 

Kinds  of  Insurance  Terms 

Insurance  involves  a  contract  guaranteeing  an  indemnity 
in  case  of  loss  resulting  from  certain  causes.  When  such  a 
contract  is  entered  into  by  a  number  of  persons  who  agree 
to  mutually  share  losses  it  is  known  as  mutual  insurance. 

When  a  company  is  organized  as  an  investment  and  the 
stockholders  agree  to  share  the  profits  and  the  losses  it  is 
known  as  a  stock  company.  Sometimes  the  principles  of 
the  Mutual  and  the  Stock  Company  are  combined  into  a 
Mixed  Company.  In  such  an  organization  all  of  the  earn- 
ings of  the  company  above  a  specified  dividend  on  the  stock 
are  divided.  The  company  assuming  an  insurance  risk  is 
called  the  Insurer  or  the  Underwriter. 

The  contract  between  the  insurer  and  the  insured  is 


262  HOW  TO  TEACH  AEITHMETIG 

called  a  policy.  It  states  the  conditions  under  which  the 
guarantee  against  loss  is  made,  the  time  the  policy  is  to 
be  in  force,  the  rate,  and  other  necessary  facts. 

If  an  insurance  company  guarantees  against  loss  of  prop- 
erty it  is  called  a  property  insurance  company.  The  prin- 
cipal kinds  of  property  insurance  are :  fire,  tornado,  light- 
ning, burglary,  live  stock,  marine,  plate  glass,  steam  boilers, 
and  transit.  Personal  insurance  secures  to  the  insured  or 
his  heirs  a  certain  sum  in  case  of  sickness,  accident  or  death. 
The  principal  kinds  of  personal  insurance  are  life,  health, 
and  accident. 

A  policy  may  be  closed  or  open.  A  closed  policy  is  one 
in  which  the  amount  of  the  indemnity  in  case  of  loss  is 
specified  in  the  policy.  An  open  policy  is  one  in  which 
the  amount  of  the  indemnity  is  to  be  determned  after  the 
loss. 

The  premium  is  the  amount  paid  for  the  insurance.  In 
mutual  companies  the  cost  of  insurance  depends  largely 
upon  the  losses  suffered  by  members  of  the  company.  In 
stock  companies  a  definite  premium  is  charged  for  "insuring 
for  a  given  time.  In  mutual  companies  the  cost  to  the 
policy  holder  is  called  an  assessment.  The  rate  of  insur- 
ance depends  upon  the  nature  of  the  risk,  the  face  of  the 
policy,  and  the  period  for  which  the  policy  is  to  be  in 
operation.  It  is  sometimes  stated  as  a  certain  per  cent 
of  the  face  of  the  policy,  but  more  frequently  it  is  quoted 
as  so  much  per  $100. 

The  rates  of  insurance  on  property  vary  with  the  kinds 
of  buildings,  their  location  with  reference  to  other  build- 
ings, the  fire  protection  in  the  community,  etc.  Thus,  the 
rate  on  a  brick  building,  other  things  being  equal,  is  less 
than  on  a  frame  structure.  The  rate  on  a  livery  barn  is 
high  because  of  the  relatively  great  likelihood  of  fire  in 
such  a  structure. 


THE  APPLICATIONS  OF  PERCENTAGE  263 

Fire  Insurance 

Fire  insurance  on  buildings  is  usually  written  for  one, 
three,  or  five  years.  Insurance  upon  the  contents  of  a 
store  or  upon  grain  is  frequently  for  a  much  shorter  period. 
Fire  insurance  usually  covers  not  only  the  loss  from  fire, 
but  also  from  smoke  and  water  and  damage  done  by 
firemen  in  putting  out  a  fire  in  adjoining  buildings.  It  is 
customary  to  insure  a  property  for  about  three-fourths  of 
its  actual  value.  Any  person  who  has  an  interest  in  a 
property  may  insure  his  interest. 

Some  fire  insurance  policies  contain  an  "average  clause. " 
When  such  is  the  case  the  company  will  in  case  of  loss  pay 
such  a  part  of  the  loss  as  the  policy  is  of  the  value  of  the 
property  insured. 

For  example :  when  property  worth  $5000  is  insured 
for  one-half  of  its  value,  or  $2500,  the  company  whose 
policy  contains  an  "average  clause"  will,  in  case  of  total 
loss,  pay  only  one-half  of  the  loss.  If  the  "average 
clause"  were  not  in  the  policy  the  entire  face  of  the  policy 
would  be  paid.  Insurance  companies  sometimes  assume  a 
risk  and  then  reinsure  a  part  of  the  risk  in  other  companies. 

Ask.  the  pupils  to  find  out  the  different  rates  on  various 
kinds  of  buildings  in  the  community.  Eequire  them  to 
account  for  the  difference  in  rates.  If  a  man  constructs  a 
building  that  is  more  nearly  fire  proof  than  other  build- 
ings of  the  same  kind  does  he  thereby  save  on  his  insur- 
ance premiums?  Should  schoolhouses  and  courthouses  be 
insured?  Why?  Why  is  a  property  usually  not  insured 
for  its  full  value  ? 

Consider  the  details  of  some  local  fire  and  show  what  the 
duties  of  an  insurance  adjuster  are.  The  teacher  should 
secure  some  fire  insurance  policies  and  these  should  be 
used  in  the  class.  As  far  as  possible  the  problems  should 
be  based  upon  local  conditions. 


264 


HOW  TO  TEACH  ARITHMETIC 


Life  Insurance 


The  three  principal  kinds  of  life  insurance  policies  are 
the  straight  life,  endowment;  and  limited  payment  life. 

The  object  and  the  principal  advantages  of  each  of  these 
types  of  policies  should  be  understood.  If  possible,  the 
teacher  should  show  the  class  a  policy  of  each  of  these 
types,  and  the  distinguishing  features  and  relative  advan- 
tages and  disadvantages  of  each  should  be  discussed.  The 
advantages  and  disadvantages  of  fraternal  or  assessment 
insurance  may  be  briefly  discussed,  and  the  chief  dif- 
ferences between  these  and  "old  line"  companies  should 
be  pointed  out. 

A  person  may  insure  his  own  life  or  that  of  any  person 
in  whom  he  has  a  pecuniary  interest,  or  upon  whom  he 
depends  for  support.  The  necessity  for  a  physical  ex- 
amination of  the  applicant  for  insurance  and  for  the 
numerous  facts  of  his  family  history  should  be  made 
evident. 

Pupils  will  be  much  interested  in  comparing  the  rate  of 
increase  in  cost  of  insurance  as  a  man  becomes  older. 
Pupils  are  especially  interested  in  a  table  which  shows  the 
expectancy  of  life. 

CARLISLE  TABLE  OF  EXPECTANCY  OF  LIFE 


Expectancy  Expectancy  Expectancy       Expectancy 

Age.      in  years.       Age.       in  years.      Age.       in  years.     Age.    in  years. 


0 

38.72 

26 

37.14 

52 

19.68 

78 

6.12 

1 

44.68 

27 

36.41 

53 

18.97 

79 

5.80 

2 

47.55 

28 

35.69 

54 

18.28 

80 

5.51 

3 

49.82 

29 

35.00 

55 

17.58 

81 

5.21 

4 

50.76 

30 

34.34 

56 

16.89 

82 

4.93 

5 

51.25 

31 

33.68 

57 

16.21 

83 

4.65 

6 

51.17 

32 

33.03 

58 

15.55 

84 

4.39 

7 

50.80 

33 

32.36 

59 

14.92 

85 

4.12 

THE  APPLICATIONS  OF  PERCENTAGE 


CARLISLE  TABLE  OF  EXPECTANCY  OF  LIFE — Continued 


2Gb 


Expectancy 
Age.   in  years. 

Expectancy 
Age.   in  years. 

Expectancy 
Age.   in  years. 

Expectancy 
Age.  in  years. 

8 

50.24 

34 

31.68 

60 

14.34 

86 

3.90 

9 

49.57 

35 

31.00 

61 

13.82 

87 

3.71 

10 

48.82 

36 

30.32 

62 

13.31 

88 

3.59 

11 

48.04 

37 

29.64 

63 

12.81 

89 

3.47 

12 

47.27 

38 

28.96 

64 

12.30 

90 

3.28 

13 

46.51 

39 

28.28 

65 

11.79 

91 

3.26 

14 

45.75 

40 

27.61 

66 

11.27 

92 

3.37 

15 

45.00 

41 

26.97 

67 

10.75 

93 

3.48 

16 

44.27  , 

42 

26.34 

68 

10.23 

94 

3.53  . 

17 

43.57 

43 

25.71 

69 

9.70 

95 

3.53 

18 

42.87 

44 

25.09 

70 

9.18 

96 

3.46 

19 

42.17 

45 

24.46 

71 

8.65 

97 

3.28 

20 

41.46 

46 

23.82 

72 

8.16 

98 

3.07 

21 

40.75 

47 

23.17 

73 

7.72 

99 

2.77 

22 

40.04 

48 

22.50 

74 

7.33 

100 

2.28 

23 

39.31 

49 

21.81 

75 

7.01 

101 

1.79 

24 

38.59 

50 

21.11 

76 

6.69 

102 

1.30 

25 

37.86 

51 

20.39 

77 

6.40 

103 

0.83 

The  expectancy  of  life  of  a  man  who  is  22  years  of  age  is  40.04 
years;  that  of  a  man  who  is  46  years  of  age  is  23.82  years. 

The  premium  charged  by  a  given  life  insurance  com- 
pany for  a  given  policy  is  determined  by  three  considera- 
tions: (1)  The  age  of  the  insured.  (2)  The  expenses  of 
managing  the  company.  (3)  The  rate  of  interest  that  the 
company  can  earn  upon  the  premium  received. 

State  laws  differ  greatly  in  the  extent  to  which  they  pro- 
tect policy  holders,  and  this  fact  should  be  mentioned  in 
discussing  the  subject  of  insurance. 

Illustrations 

Suppose  that  Mr.  A,  who  is  25  years  old,  takes  out  an 
ordinary  life  policy  for  $1000,  for  which  he  pays  an  annual 
premium  of  $21.35.  After  paying  four  annual  premiums, 


266  HOW  TO  TEACH  ARITHMETIC 

or  $85.40,  A  dies  and  the  company  pays  his  beneficiary 
$1000. 

Pupils  may  be  puzzled  to  know  how  the  company  can 
afford  to  do  this.  The  expectation  of  life  for  a  man  25 
years  of  age  is  37.86  years,  gad  many  who  take  insurance 
when  they  are  25  years  of  age  live  more  than  37.86  years. 
There  are  many  who  take  out  a  policy  and  after  car- 
rying it  for  a  few  years  drop  it  and  so  receive  nothing. 
If  A  had  lived  50  years  after  taking  out  his  policy  and  had 
paid  his  annual  premiums  he  would  have  paid  in  $1067.50. 
The  insurance  company  would  have  put  each  sum  on  inter- 
est when  it  was  paid  and  the  entire  sum  at  compound  inter- 
est would  probably  have  amounted  to  more  than  $2500.  A 
might  have  invested  his  money  at  a  higher  rate  than  the 
insurance  company  did,  but  he  was  willing  to  pay  $21.35 
annually  to  an  insurance  company  for  assuming  the  risk  on 
his  life. 

Similar  considerations  may  be  discussed  for  the  endow- 
ment and  the  limited  payment  life  policies.  The  reason 
why  the  limited  payment  life  costs  more  than  the  straight 
life  and  why  the  endowment  costs  more  than  the  other  two 
should  be  clear  to  the  pupil. 

The  text-book  problems  on  insurance  involve  no  mathe- 
matical difficulties,  and  if  the  pupil  understands  the  gen- 
eral features  of  the  subject  he  should  have  no  difficulty  in 
solving  the  problems. 

TAXES  AND  REVENUE 

Unlike  some  of  the  applications  of  percentage,  the  sub- 
ject of  taxes  is  one  in  which  most  of  the  homes  represented 
by  the  pupils  are  directly  interested.  Once  or  twice  each 
year  the  payment  of  taxes  becomes  a  very  real  matter 
and  the  entire  subject  can  be  made  interesting  to  the 


THE  APPLICATIONS  OF  PERCENTAGE  267 

pupils  if  most  of  the  problems  are  based  on  local  condi- 
tions. A  detailed  discussion  of  the  subject  is  a  part  of 
civics  rather  than  of  arithmetic. 


Why  Taxes  Are  Necessary 

The  first  essential  is  that  the  pupils  should  understand 
why  the  levying  and  collecting  of  taxes  is  necessary.  -It 
should  be  explained  that  a  tax  is  any  money  levied  by  a 
government, — national,  state,  county,  city, — to  defray  all 
or  a  part  of  its  expenses.  Pupils  should  be  asked  to  name 
several  items  of  expense  "of  the  nation,  state,  county,  town- 
ship, and  city;  such  as  the  salaries  of  all  state  officials, 
erection  and  maintenance  of  public  hospitals,  asylums 
and  prisons,  state  schools  and  public  buildings.  The 
county  must  raise  money  to  defray  the  expense  incurred 
in  paying  the  salaries  of  county  officials,  the  erection  and 
maintenance  of  the  court  house  and  other  county  build- 
ings, the  building  and  repairing  of  roads,  etc.  The  city 
must  levy  a  tax  to  maintain  the  public  schools,  to  pro- 
vide proper  sewerage  and  other  sanitary  conditions,  to 
provide  police  and  fire  protection,  to  pay  the  salaries  of 
city  officials,  etc.  The  smaller  units  of  government  must 
levy  tax  to  build  and  repair  roads  and  bridges  and  to 
defray  other  items  of  public  expense.  Ask  the  pupils  to 
compare  the  taxes  which  a  man  pays  each  year  with  what 
he  would  have  to  pay  to  secure  equivalent  service  and  pro- 
tection if  no  taxes  were  levied.  If  taxes  were  not  levied  he 
would  have  to  pay  for  the  education  of  his  own  children, 
and  this  item  alone  would,  in  most  cases,  be  larger  than  his 
annual  taxes.  He  would  also  need  to  provide  for  protec- 
tion from  fire  and  theft.  The  pupils  should  see  that  a  just 
levy  of  taxes  is  not  only  wise  but  economical  for  the 
individual  citizen. 


268  HOW  TO  TEACH  AEITHMETIC 

Kinds  of  Tax 

For  purposes  of  taxation  there  are  two  kinds  of  prop- 
erty tax, — real  estate  and  personal.  Real  estate  comprises 
property  that  is  not  easily  moved,  such  as  land,  mines, 
quarries,  buildings,  railroads,  and  forests.  Personal  prop- 
erty is  easily  moved,  such  as  money,  stocks,  bonds,  live 
stock,  and  household  goods.  Separate  assessments  are 
made  for  each  kind  of  property,  and  in  some  localities 
the  personal  and  real  taxes  are  due  on  different  dates. 
Pupils  should  be  asked  to  explain  why  the  rates  of  city 
and  county  taxes  in  a  given  locality  differ.  "What  pro- 
tection or  convenience  does  the  man  who  resides  in  the 
city  have  that  his  friend  in  the  country  does  not  have? 
Is  it  reasonable  that  the  resident  of  the  city  should  be 
assessed  for  this  protection  and  convenience  1 

Besides  the  property  tax  just  mentioned  there  are  sev- 
eral other  kinds  of  tax.  In  many  states  a  poll  tax  of  a 
specified  sum  is  levied  on  all  male  citizens  over  twenty- 
one  years  of  age.  Franchises  which  are  granted  by  munici- 
palities or  counties  are  often  subject  to  a  special  tax.  In 
many  localities  a  special  dog  tax  is  levied.  A  special  tax 
is  also  levied  in  some  states  on  all  inheritances  over  a  cer- 
tain prescribed  minimum. 

Method  of  Levying  and  Collecting  Taxes 

The  method  of  levying  and  collecting  taxes  varies  in  the 
different  states  and  a  detailed  consideration  of  the  various 
methods  would  be  too  long  for  this  book,  but  the  teacher 
should  not  fail  to  ascertain  the  method  used  in  the  state 
and  in  the  county  in  which  he  is  teaching  and  to  familiarize 
the  pupil  with  it.  Such  a  study  should  acquaint  the  pupil 
with  the  meaning  and  use  of  the  following  terms :  valua- 


THE  APPLICATIONS  OF  PEECENTAGE  269 

tion,  assessment,  assessor,  board  of  review,  board  of  equali- 
zation, tax  collector,  and  delinquent  taxes. 

How  Tax  Hates  Are  Determined 

The  method  by  which  the  rate  for  a  given  kind  of  tax 
is  determined  should  be  clearly  understood.  The  state 
legislature  determines  the  amount  to  be  spent  by  the  state. 
The  assessed  value  of  the  property  of  the  state  is  ascer- 
tained and  from  these  two  items  the  rate  of  the  state  tax 
is  determined.  For  illustration:  if  the  state  legislature 
authorizes  the  expenditure  of  $18,000,000  and  the  prop- 
erty valuation  in  the  state  is  $3,000,000,000,  the  rate  of  the 
state  tax  will  be  .006.  Tax  rates  are  usually  expressed  as 
so  many  mills  on  each  dollar  of  valuation.  In  the  illus- 
tration just  given  the  state  tax  would  be  6  mills  on  each 
dollar.  The  rate  for  school,  road  and  bridge  and  other 
taxes  in  a  community  is  determined  in  the  same  general 
manner.  The  pupil  should  be  told  what  body  is  authorized 
to  determine  the  amount  of  each  of  the  above  taxes. 

In  many  localities  it  is  the  custom  to  assess  property  at 
\  or  f  of  its  real  value. 

The  time  and  labor  of  computing  taxes  are  reduced  by 
the  use  of  tables  from  which  the  tax  on  various  sums  at 
specified  rates  may  be  easily  found. 

Base  Problems  on  Local  Conditions 

Most  teachers  can  obtain  an  assessment  blank  and  several 
tax  receipts  for  class  use.  Problems  based  on  local  levies 
and  assessments  are  of  much  more  interest  to  the  pupil 
than  most  of  the  problems  of  the  text-book.  Details  in  re- 
gard to  state  revenue  laws  may  be  secured  by  addressing 
the  secretary  of  state. 


270  HOW  TO  TEACH  AKITHMETIC 

Duties  and  Revenues 

The  subject  of  national  duties  and  revenues  is  closely 
related  to  that  of  taxes  and  may  be  presented  in  a  similar 
manner.  Citizens*  are  not  taxed  directly  for  the  support 
of  the  national  government.  Among  the  chief  items  of  na- 
tional expense  may  be  mentioned  the  maintenance  of  the 
Army  and  the  Navy,  Diplomatic  and  Consular  Service. 
Indians,  Internal  Improvements,  Interest  on  the  Public 
Debt,  Erection  and  Maintenance  of  Public  Buildings,  Sal- 
aries of  government  officials,  Pensions,  etc.  The  expenses  of 
the  National  Government  are  defrayed  by  the  receipts  from 
duties  and  customs,  internal  revenues  (liquors,  tobacco, 
etc.),  sale  of  public  lands,  income  and  corporation  taxes,  etc. 

The  study  of  revenues  introduces  the  question  of  specific 
and  ad  valorem  duties,  the  tariff  and  kindred  subjects. 
The  teacher  must  use  his  discretion  as  to  how  fully  these 
subjects  should  be  considered  in  a  class.  The  mathematics 
involved  in  the  problems  of  taxes  and  national  revenues 
is  so  simple  that  illustrative  problems  are  not  necessary. 
The  entire  subject  should  acquaint  the  pupil  with  local 
and  national  methods  of  defraying  necessary  expenses. 
This  involves  a  study  of  the  elements  of  civics  and 
of  economics  rather  than  of  arithmetic.  The  mathematics 
involved  should  be  considered  as  subordinate  to  the  other 
phases  of  the  subject. 


CHAPTER  XVII 

BANKING;  CORPORATIONS;  BUSINESS  PRACTICE 

The  modern  bank  is  a  complex  institution;  it  is  not 
the  duty  of  the  school  to  acquaint  pupils  with  all  of  the 
technicalities  involved  in  banking.  Pupils  should  know 
the  function  of  a  bank  in  modern  commercial  life ;  the 
principal  duties  of  the  book-keepers  and  the  cashiers ;  how 
to  deposit  money  and  to  make  out  and  endorse  checks. 

The  Chief  Functions  of  a  Bank 

1.  To  receive  money  on  deposit  and  pay  checks  drawn 

on  deposits. 

2.  To  lend  money  on  promissory  notes. 

3.  To  sell  drafts  on  other  banks  and  collect  drafts  on 

persons. 

4.  To  issue  its  own  promissory  notes,  which  serve  as 

currency    (only  National  Banks  can  do  this  in 
the  United  States.) 

The  School  Bank 

Problems  involving  banking  should  be  made  realistic  to 
the  pupils.  The  nearer  the  approach  to  actual  business 
the  greater  will  be  the  interest  of  many  of  the  pupils. 
The  school  may  have  a  bank  in  which  certain  pupils  act 
as  cashier,  paying  teller,  receiving  teller,  bookkeeper,  etc. 
Inexpensive  paper  money,  deposit  slips,  and  checks  may 
be  secured.  More  than  ninety  per  cent  of  the  business  of 
the  country  is  carried  on  by  checks  and  drafts,  but  few 
pupils  have  seen  either.  In  some  localities  it  will  be  pos- 

271 


272  HOW  TO  TEACH  ARITHMETIC 

sible  to  induce  a  local  banker  to  talk  to  the  pupils  on  such 
subjects  as  the  clearing  house. 

Negotiable  Paper 

Pupils  should  become  familiar  with  the  principal  forms 
of  promissory  notes,  bills  of  exchange,  and  checks.  These 
instruments  are  an  important  factor  in  business  trans- 
actions, passing  from  hand  to  hand  as  a  substitute  for 
money. 

No  exact  form  need  be  followed  in  order  to  make  a 
contract  a  negotiable  instrument,  but  custom  has  pre- 
scribed forms  that  are  very  generally  used.  A  negotiable 
instrument,  whether  it  is  a  promissory  note,  bill  of  ex- 
change or  check,  must  be: 

(a)  In  writing.     (No  oral  contract  is  negotiable). 

(b)  Properly  signed.     (The  name  is  not  necessary;  any 
mark  intended  to  serve  as  a  signature  will  suffice). 

(c)  Negotiable  in  form.      (If  made  payable  to  a  par- 
ticular person  or  persons  it  is  not  negotiable). 

(d)  Payable  in  money  only.     (The  value  is  then  definite 
and  certain). 

(e)  The  amount  must  be  definitely  stated. 

(f)  It  must  be  payable  absolutely.     (Not  upon  certain 
conditions) . 

(g)  To  the  order  of  a  designated  payee  or  bearer.     (To 
a  person  or  persons  who  can  be  identified  when  the  note 
matures). 

(h)  At  a  certain  time.  (Not  at  a  time  contingent  upon 
some  other  event,  but  at  a  time  sure  to  arise). 

Promissory  Notes 

The  party  who  makes  the  note  and  whose  promise  is 
Stated  is  called  the  maker.  The  party  to  whom  the  promise 


BANKING;  CORPORATIONS;  BUSINESS  PEACTICE       273 

is  made  is  called  the  payee.  If  the  note  is  without  interest 
this  fact  may  be  stated  in  the  note  or  the  words  "with 
interest"  may  be  omitted. 

The  payee  of  a  negotiable  note  may  transfer  his  rights 
to  another  by  writing  his  name  upon  the  back  of  the  note. 
This  is  called  an  indorsement,  and  the  person  to  whom  the 
note  is  transferred  is  called  the  indorsee. 
$1800.  Urbana,  Illinois, 

September  28,  1913. 

Sixty  days  after  date  I  promise  to  pay  to  the  order  of 
James  Riley,  Eighteen  Hundred  Dollars,  without  inter- 
est. Value  received. 

JOHN  BAKER. 

Ask  the  pupil  who  is  the  maker  of  the  above  note  ?  "Who 
is  the  payee  ?  When  is  the  note  due  ?  Who  pays  the  note 
when  it  matures?  To  whom?  How  much  must  be  paid 
when  the  note  matures?  Who  gets  the  check  after  it  is 
paid?  Why  are  the  words  "value  received"  put  in  a 
note?  Is  the  above  note  negotiable?  Suppose  the  payee 
wishes  to  sell  the  note  before  it  matures;  how  should  he 
indorse  it? 

The  endorsement  in  the  illustration  is  called  an  endorse- 
ment in  full.  If  the  note  is  endorsed  by  the  words  "pay 
to  William  Jones  only"  it  cannot  be  transferred  again. 
This  is  called  a  "restricted  endorsement."  The  holder  of 
the  note  may  simply  write  his  name  across  the  back  of  it. 
This  is  called  an  endorsement  in  blank,  and  makes  the  note 
payable  to  bearer.  The  question  of  liability  incurred  by 
the  indorser  of  a  note  should  be  discussed.  The  extent  to 
which  the  indorser  of  a  note  limits  his  liability  by  using  the 
words  "without  recourse"  should  be  considered. 

There  are  numerous  kinds  of  notes  and  drafts  involving 
more  or  less  technicalities.  The  teacher  should  acquaint 
the  pupils  with  only  the  simpler  forms  in  common  use.  In 


274 


HOW  TO  TEACH  ARITHMETIC 


Pay  to  the  order  of 
WILLIAM  JONES. 
JAMES  RILEY. 


treating  the  subject  of  negotiable  paper  a  good  commercial 

arithmetic  will  be  of  value  to  the  teacher. 

When  money  is  borrowed 
from  a  bank  the  security 
given  may  be  (a)  real  es- 
tate, (b)  collateral,  (c) 
'personal.  These  should  be 
discussed  in  the  class  and 
given  practical  setting  by 
use  in  the  school  bank. 

Discounting  Notes 

Suppose  that  James  Riley, 
the  payee  of  the  note  on 
page  273,  sells  the  note  to 
a  bank  on  October  23.  It 
is  evident  that  the  bank 
could  not  afford  to  pay 
$1800  for  it  because  that 
amount  will  not  be  paid  to 
the  holder  of  the  note  until 
it  is  due, — November  27; 
the  bank  would  lose  the 
interest  on  $1800  from 
October  23  to  November  27. 
The  interval  between  the  date  a  note  is  discounted  and  the 
date  of  maturity  is  called  "the  term  of  discount.'7  The 
difference  between  the  value  of  a  note  at  maturity  and  the 
amount  paid  for  the  note  when  it  is  sold  is  called  discount. 
A  note  that  does  not  bear  interest  matures  at  its  face 
value.  If  it  is  interest-bearing  from  date,  the  interest 
which  will  have  been  earned  at  the  date  of  maturity  must 
be  added  to  the  face  of  the  note  to  determine  its  maturity 
value. 


BANKING;  COKPOKATIONS;  BUSINESS  PEACTICE       275 

The  first  step  in  calculating  discount  on  a  note  is  to  find 
the  maturity  value  since  the  purchaser  must  know  this 
before  he  can  determine  how  much  he  can  afford  to  pay 
for  it. 

In  computing  the  term  of  discount  banks  usually  count 
the  exact  number  of  days.  In  some  states  days  of  grace 
are  still  permitted.  If  a  note  is  due  a  certain  number  of 
months  after  date  it  will  mature  on  the  same  day  of  the 
month  as  the  date  of  the  note. 

For  convenience  the  following  table  is  often  used  by 
bankers  to  find  the  days  intervening  between  dates. 

DISCOUNT  TABLE 
To  the  same  day  of  the  next 

JH  *J          g          >g^H^t>OQ          O         £         W 

<t>  H£  ££d^<^oct> 

£         cr1        £         ^       ^        £       ^7*      fro       ^3         ff"       3      . « 

TT'.-~,~.     «_~  rt  M  fcj.^/T*  ^<  yT  r-t-  O  _ !  (Ti 


±rom  any 
day   . 

P 

q 

a 

pa 

43 

t- 

w 

1 

O 

i 

cr1 
as 
i-i 

cr1 
? 

cr 
? 

•-i 

January  365  31  59  90  120  151  181  212  243  273  304  334 

February  33  365  28  59  89  120  150  181  212  242  273  303 

March   306  337  365  31  61  92  122  153  184  214  245  275 

April  275  306  334  365  30  61  91  122  153  183  214  244 

May  245  276  304  335  365  31  61  92  122  153  184  214 

June  214  245  273  304  334  365  30  61  92  122  153  183 

July  184  215  243  274  304  335  365  31  62  92  123  153 

August  153  184  212  243  273  304  334  365  31  61  92  122 

September  122  153  181  212  242  273  303  334  365  30  61  91 

October  92  123  151  182  212  243  273  304  335  365  31  61 

November  61  92  120  151  181  212  242  273  304  334  365  30 

December  31  62  90  121  151  182  212  243  274  304  335  365 

The  exact  number  of  days  from  any  day  of  one  month 
to  the  same  day  of  another  month,  within  a  year  is  found 
by  starting  at  the  name  of  the  first  month  in  the  left 
hand  column  and  following  across  to  the  column  of  the 
last  named  month.  Thus  to  find  the  number  of  days  from 


276  HOW  TO  TEACH  ARITHMETIC 

April  18  to  August  24,  we  note  that  it  is  122  days  from 
April  18  to  August  18,  and  then  we  add  6  days  (August 
18  to  August  24)  which  gives  128  days  as  the  exact  time. 

Suppose  we  wish  to  find  the  bank  discount  on  the  note 
given  on  page  273,  if  the  note  is  discounted  on  October  23 
at  6%. 

$1800  =  maturity  value  of  the  note.     (Future  worth). 

Time  from  October  22  to  November  27  is  36  days,  this 
is  the  term  of  discount. 

Interest  on  $1800  for  36  days  at  6%  is  $10.80,  this  is 
the  bank  discount. 

The  seller  of  the  note  would  receive  $1800  -  $10.80,  which 
is  $1789.20. 

If  the  preceding  note  had  been  drawn  with  interest  at 
6%  the  maturity  value  of  the  note  would  have  been  $1800 
plus  the  interest  on  $1800  at  6%  for  the  time  of  the  note 
(60  days.)  This  would  have  been  the  basis  for  computing 
the  bank  discount. 

Partial  Payments 

The  subject  of  partial  payments  is  no  longer  of  great 
value  because  of  changes  that  have  been  made  in  the  form 
of  notes.  The  modern  bank  is  rendering  the  subject  obso- 
lete. The  numerous  state  rules  should  not  be  taught. 

Methods  of  Transmitting  Money 

Seme  attention  should  be  devoted  to  the  various  ways  of 
sending  money  from  one  locality  to  another.  The  advan- 
tages and  disadvantages  of  each  should  be  briefly  discussed. 

Debts  between  men  in  the  same  community  are  usually 
paid  by  means  of  checks  or  of  actual  currency.  Debts 
between  persons  in  different  communities  may  be  dis- 
charged by  any  of  the  means  enumerated  below.: 


BANKING;  CORPORATIONS;  BUSINESS  PRACTICE       277 

1.  Sending  actual  currency  or  stamps. 

2.  Check. 

3.  Bank  draft. 

4.  Registered  letter. 

5.  Postal  money  order. 

6.  Express  money  order. 

7.  Telegraph  money  orders. 

Very  little  attention  should  be  given  to  the  technicalities 
involved  in  the  subject  of  exchange. 

Bookkeeping 

Pupils  should  be  taught  how  to  keep  simple  accounts. 
The  average  man  or  woman  finds  but  little  occasion  for 
any  elaborate  form.  Most  business  men  prefer  to  teach 
their  own  method  to  one  who  will  have  charge  of  their 
books. 

Pupils  should,  however,  be  taught  how  to  keep  a  neat 
and  accurate  record  of  the  expenses  of  the  home,  the  farm, 
the  store,  and  the  like. 

Stocks  and  Bonds 

The  subject  of  stocks  and  bonds  is  generally  considered 
to  be  the  most  difficult  application  of  percentage.  The  diffi- 
culty is  due,  however,  not  to  the  mathematics  involved,  but 
to  the  fact  that  the  language  of  the  subject  is,  frequently 
unfamiliar  to  the  teacher  and  is  usually  unfamiliar  to  the 
pupil. 

Organization  of  a  Corporation 

The  best  way  to  approach  the  subject  in  order  to  make 
it  real  to  the  pupils  is  to  let  the  pupils  organize  a  corpora- 
tion. When  such  an  organization  is  completed  the  pupil 
will  have  a  better  knowledge  of  the  meaning  of  the  techni- 


278  HOW  TO  TEACH  ARITHMETIC 

cal  terms  of  stocks  and  bonds  than  he  would  have  had 
from  a  study  of  the  terms  in  a  more  formal  manner.  Cor- 
porations are  assuming  control  of  many  business  enterprises 
and  are  replacing  the  individual  in  larger  business  ventures. 

Before  a  corporation  can  be  organized  it  must  secure  a 
commission  from  the  secretary  of  state.  The  persons  com- 
missioned by  the  secretary  of  state  are  permitted  to  take 
subscriptions  for  the  stock.  After  the  stock  is  subscribed 
a  meeting  is  held  for  the  purpose  of  effecting  a  temporary 
organization.  A  constitution  is  adopted  and  the  officers 
provided  for  in  the  constitution  are  elected.  The  name  and 
purpose  of  the  organization;  its  location,  the  number  of 
shares  and  par  value  of  each  share,  the  officers  and  the  time 
for  holding  regular  meetings  of  the  stockholders  are  deter- 
mined upon  and  a  statement  of  these  facts  is  filed  with  the 
secretary  of  state.  A  charter  is  then  issued  and  the  or- 
ganization is  completed.  The  stock  previously  subscribed 
for  is  sold.  The  directors  are  empowered  to  buy,  sell,  con- 
tract debts,  etc. 

Suppose  that  the  pupils  organize  a  company  with  a  capi- 
tal stock  of  $100,000,  the  par  value  of  each  share  being 
$100.  (The  par  value  may  be  $500  or  $50,  or  $10,  or  any 
other  convenient  amount.)  The  pupil  who  buys  20  shares 
of  the  stock  will  pay  $2000  for  it,  if  the  stock  sells  at  par 
value.  For  each  share  that  one  owns  he  is  entitled  to  one 
vote  at  all  meetings  of  the  stockholders.  When  one  pays 
for  his  stock  he  receives  a  "certificate  of  stock/'  stating 
the  number  of  shares  bought,  the  par  value  of  each  share, 
etc.  The  owner  of  stock  is  entitled  to  participate  in  the 
profits  in  proportion  to  the  number  of  shares  owned. 

Corporations  often  issue  both  preferred  and  common 
stock.  A  preferred  stock  certificate  states  that  dividends 
up  to  a  certain  limit,  usually  4%  to  7%,  are  to  be  paid 
before  any  dividends  are  paid  on  common  stock.  Holders 


BANKING;  CORPORATIONS;  BUSINESS  PRACTICE       279 

of  preferred  stock  have  first  chance  at  dividends,  but  their 
dividends  are  limited. 

Suppose  that  at  the  end  of  the  first  year  the  corporation 
organized  by  the  pupils  has  cleared  $12,000  above  all  ex- 
penses. The  directors  may  decide  to  declare  this  as  a 
dividend,  or  all  or  a  part  of  it  may  be  put  into  a  "  reserve 
fund. ' '  If  it  is  declared  as  a  dividend  and  if  the  holders 
of  preferred  stock  are  entitled  to  6%,  this  must  be  paid 
before  any  dividend  is  computed  for  the  holders  of  com- 
mon stock.  Let  us  suppose  that  $30,000  of  the  entire  stock 
($100,000)  is  preferred  stock.  The  holders  of  this  stock 
are  entitled  to  6%  of  $30,000,  or  $1800.  A  dividend  of 
$6  is  paid  on  each  share.  The  holders  of  the  common  stock 
are  entitled  to  the  balance  of  earnings,  or  $10,200.  This 
dividend  will  be  divided  among  the  holders  of  $70,000 
worth  of  stock.  Each  holder  of  common  stock  will,  there- 
fore receive  14f%  dividend  on  each  share,  since  this  is 
higher  than  the  current  rate  of  interest,  the  common  stock 
will  sell  above  par,  that  is,  at  a  premium. 

Suppose  that  at  the  end  of  the  second  year  the  corpora- 
tion has  cleared  but  $1800.  If  dividends  are  declared  the 
holders  of  preferred  stock  are  entitled  to  this  entire  amount, 
since  it  is  just  6%  of  the  face  value  of  the  preferred  stock. 
The  holders  of  common  stock  would  receive  no  dividends 
and  their  stock  would  likely  sell  below  par.  In  case  the  net 
earnings  of  the  corporation  are  not  sufficient  to  pay  6% 
dividends  on  the  preferred  stock,  the  earnings  of  the  cor- 
poration, if  declared  as  dividends,  will  be  divided  pro  rata 
among  the  holders  of  preferred  stock. 

Bonds 

/ 

Corporations  may  borrow  money,  pledging  their  prop- 
erty as  security  in  the  same  way  as  individuals.  When  a 


280  HOW  TO  TEACH  ARITHMETIC 

large  amount  of  money  is  to  be  raised  it  is  usually  done  by 
issuing  bonds.  A  bond  is  a  mortgage  note,  upon  which 
the  corporation  agrees  to  pay  interest,  and  to  the  payment 
of  which  the  entire  property  and  business  of  the  corpora- 
tion is  pledged. 

Bonds  may  be  issued  by  national  and  state  governments, 
counties,  townships,  cities,  school  districts,  etc.  They  are 
declared  by  law  to  be  "bodies  corporate." 

Bonds  made  payable  to  the  owner  or  his  assignee  are 
called  registered  bonds;  bonds  made  payable  to  bearer  are 
termed  coupon  bonds.  In  such  bonds  the  interest  is  pro- 
vided for  in  attached  notes  or  "coupons."  In  the  case 
of  registered  bonds  the  interest  is  sent  directly  to  the 
owner.  The  holder  of  a  coupon  bond  must  surrender  a 
coupon  when  each  interest  payment  is  made. 

If  a  bond  pays  more  than  the  current  rate  of  interest 
and  is  safe,  it  usually  sells  at  a  premium. 

The  price  .of  many  stocks  and  bonds  is  quoted  in  the 
daily  papers.  Market  values  often  fluctuate  widely  and 
an  intimate  knowledge  of  corporations  and  market  con- 
ditions is  necessary  if  one  expects  to  invest  wisely.  A 
broker  is  assumed  to  know  the  stocks  and  bond  market 
and  to  be  able  to  advise  his  clients  when  to  buy  or  to  sell 
stocks  and  bonds.  The  fee  paid  him  for  his  services  is 
called  commission  or  brokerage,  it  is  usually  about  £%. 
It  should  be  remembered  that  brokerage  is  always  a  per 
cent  of  the  par  value  whether  stock  and  bonds  are  bought 
or  sold.  The  technicalities  of  the  broker's  office  are  not  of 
value  to  most  citizens  and  need  not  be  taught  in  the 
schools. 

Illustrative  Problems 

Many  of  the  problems  in  stocks  and  bonds  should  be 
based  upon  the  market  quotations  given  in  the  daily  papers. 


BANKING;   CORPORATIONS;   BUSINESS  PRACTICE       281 

Illustrative  problems: 

A  broker  sells  160  shares  of  stock  at  90;  brokerage  %%. 
What  should  his  principal  receive  ? 

1.  $90  -$£  =  proceeds  from  each  share. 

160 x $89.875  =  $14,380 -proceeds  from  160  shares. 

2.  What  sum  must  be  invested  in   5%   bonds  at   110, 
without  brokerage,  to  yield  an  annual  income  of  $450? 

Each  bond  yields  $5  dividend  annually  (5%  of  par 
value). 

To  yield  $450  the  number  of  bonds  must  be  $450  -f  $5, , 
which  is  90. 

Since  each  bond  cost  $110,  the  entire  cost  was  90  x  $110 
=  $9900. 

3.  What  per  cent  does  an  investor  make  on  stock  that 
pays   a   dividend   of   5%    if  he   buys   it   at   80,   without 
brokerage  ? 

$5  =  income  on  one  share. 

$80  =  cost  of  one  share. 

The  return  on  the  investment  is  whatever  per  cent  $5 
(the  dividend)  is  of  $80  (the  amount  invested.)  $5  is 
6i%  of  $80. 

4.  What  is  the  quotation  price  of  7%  stock  that  yields 
an  income  of  10%,  no  brokerage? 

$7  =  income  on  each  share. 

$7  =  10%  of  the  amount  invested  in  one  share,  there- 
fore, the  amount  invested  is  $70. 


CHAPTER  XVIII 

THE  METEIC  SYSTEM 

Essentials  of  a  Perfect  System 

Any  perfect  system  of  weights  and  measures  must  have 
two  characteristics.  The  units  must  be  accurately  defined 
and  invariable  and  there  must  be  a  uniform  ratio  between 
consecutive  units  in  any  given  table.  The  early  systems 
of  weights  and  measures  had  neither  of  these  essentials. 
The  units  were  not  accurately  defined  and  the  scale  of 
relation  was  variable.  The  necessity  for  accurately  defined 
units  became  more  imperative  as  commerce  and  industry 
developed,  and  eventually  the  units  were  defined  with 
scientific  accuracy. 

Science  improved  the  old  system  of  weights  and  meas- 
ures by  imparting  to  its  units  the  exactness  and  uniformity 
which  is  the  first  essential,  but  it  did  not  establish  the  sim- 
ple scale  of  relationship  between  the  units,  which  is  the 
second  essential. 

The  lack  of  uniformity  in  the  ratio  between  consecutive 
units  of  the  English  system  may  be  illustrated  by  a  con- 
sideration of  the  following  tables : 

12  inches  =  1  foot 

3  feet  =  l  yard 

5J  yards  (or  16|  feet)  =1  rod 
320  rods  (or  5280  feet)  =1  mile 

16  ounces  =  1  pound 
100  pounds  =  1  hundred  weight 
20  hundred  weight  =  1  ton 
282 


THE  METRIC  SYSTEM  283 

Not  only  is  there  a  lack  of  uniformity  of  ratio  between  the 
consecutive  units  throughout  a  given  table,  but  the  same 
ratio  is  rarely  maintained  for  three  consecutive  units.  In 
linear  measure  the  numbers  expressing  the  relationship  be- 
tween consecutive  units  are  12,  3,  5^,  and  320.  In  surface 
measure  the  ratios  are  144,  9,  30^,  160,  and  640. 

In  avoirdupois  weight  the  ratios  are  16,  100,  and  20. 

It  is  evident  that  the  English  system  of  weights  and 
measures  is  lacking  in  the  second  essential  of  a  perfect 
system, — a  uniform  scale  of  relationship  between  the  con- 
secutive units  of  a  given  table. 

Origin  of  Metric  System 

The  Metric  system  is  the  result  of  an  attempt  by  the 
people  of  France  to  devise  a  perfect  system  of  weights  and 
measures, — a  system  that  would  have  both  of  the  essen- 
tials that  have  been  mentioned.  It  was  first  suggested  in 
1528,  but  it  was  not  worked  out  until  1790.  The  Metric 
system  was  the  product  of  the  minds  of  five  of  the  most 
eminent  mathematicians  of  France.  Its  history  is  very 
interesting  and  will  repay  study.  The  name  of  the  system 
is  derived  from  the  French  word  metre,  meaning  "to 
measure. ' '  The  meter  was  defined  to  be  one  ten-millionth 
of  the  distance  from  the  equator  to  the  pole,  and  this  was 
taken  as  the  standard  of  linear  measure.  All  other  units 
of  the  system  were  correlated  directly  with  the  meter.  It 
was  later  discovered  that  a  slight  mistake  was  made  in 
computing  the  length  of  the  meter,  but  this  error  does  not 
render  the  system  less  valuable. 

Present  Status  of  Metric  System 

The  enlightened  judgment  of  educated  people  pronounces 
the  system  to  be  the  best  the  world  has  yet  seen.  It  is 


284  HOW  TO  TEACH  ARITHMETIC 

endorsed  with  enthusiasm  by  those  who  have  studied  it, 
and  many  predict  that  it  will  ultimately  triumph  over  the 
English  system  and  will  become  universal  among  civilized 
nations.  The  persistence  of  custom  will  delay  its  ulti- 
mate triumph  for  many  years,  but  the  best  must  eventually 
triumph.  Since  its  adoption  in  France  in  1840  it  has 
spread  with  great  rapidity.  It  is  to-day  either  obligatory 
or  permissive  in  every  civilized  country  of  the  world.  It  is 
in  general  use  among  all  civilized  nations  except  England 
and  the  United  States.  Instruction  in  it  is  obligatory  in 
all  English  schools.  It  has  been  the  legal  system  in  the 
United  States  since  1866;  its  use  is  required  in  some  of 
the  departments  of  our  government,  and  is  authorized  in 
others. 

The  Future  of  the  Metric  System 

We  live  in  an  age  of  international  activities.  The  tend- 
encies of  the  day  are  towards  uniformity  of  language  and 
customs.  Commerce  demands  that  articles  for  export  and 
for  import  shall  be  measured  by  units  that  are  universally 
understood  among  civilized  nations.  The  complex  business 
activities  of  the  day  make  the  use  of  a  simple  system 
imperative.  One  advocate  of  the  metric  system  calls  atten- 
tion to  the  fact  that  "  there  is  no  nation  to-day  having  a 
decimal  system  of  weights  and  measures  or  currency  where 
there  is  a  suggestion  of  a  change,  and  no  nation  not  having 
a  decimal  system  where  there  is  not  constant  agitation  in 
favor  of  a  decimal  system.  Whenever  a  civilized  nation 
has  changed  its  units  the  change  has  always  been  decimal. ' ' i 
Another  ardent  advocate  of  the  metric  system  states  the 
following*:  "Considered  merely  as  a  labor-saving  machine 

i  Hon.1  James  E.  Southard,  " School,  Science  and  Mathematics," 
1905,  pp.  653-657. 


THE  METEIC  SYSTEM  285 

it  is  a  new  power  offered  to  man  incomparably  greater 
than  that  which  he  has  acquired  by  the  agency  which  he 
has  given  to  steam.  It  is  in  design  the  greatest  invention 
of  human  ingenuity  since  that  of  printing. ' ' 

There  are  two  objections  advanced  to  the  general  adop- 
tion of  the  metric  system  in  the  United  States.  The 
first  is  a  real  objection,  but  it  is  not  so  formidable  as  is 
supposed;  i.e.,  the  cost  of  changing  tools  to  correspond 
to  the  metric  units  and  the  training  of  the  laborer  to  think 
and  work  in  terms  of  these  units.  A  sudden  change  to  the 
metric  system  would  be  attended  with  confusion,  but  no 
such  change  will  take  place,  because  the  adoption  of  the 
new  units  will  be  made  gradually.  Our  money  system  is 
now  on  a  decimal  basis.  In  recent  years  several  of  the 
large  manufacturing  concerns  in  the  United  States  have 
adopted  the  metric  units  for  exclusive  use  in  shops,  and 
the  management  report  little  or  no  confusion  as  a  result 
of  the  change.  The  second  great  objection  urged  against 
the  adoption  of  the  metric  system  is  the  fact  that  in  the 
matter  of  weights  and  measures  we  are  strongly  bound  by 
the  chains  of  tradition.  The  chain  of  tradition  is  weaken- 
ing, however,  and  the  metric  system  is  gaining  ardent 
advocates  at  the  expense  of  the  English  system. 

It  is  desirable  that  teachers  should  appreciate  the  merits 
of  the  metric  system.  No  teacher  can  afford  to  be  ignorant 
of  a  system  that  has  been  so  generally  adopted  by  civilized 
nations  and  is  the  basis  for  all  scientific  work.  A  well- 
educated  person  to-day  should  know  the  metric  units  in 
common  use.  The  English  system  should  receive  much 
greater  emphasis  than  the  metric  system,  for  it  is  the  system 
that  most  pupils  will  use  in  their  daily  activities. 


2Hallock  and  Wade,  "Evolution  of  Weights  and  Measures  and  the 
Metric  System, »  pp.  116-118. 


286.  HOW  TO  TEACH  AEITHMETIC 

Where  the  Metric  System  Should  Be  Taught 

The  metric  system  should  usually  be  taught  in  the 
seventh  or  eighth  grades,  and  its  superiority  should  be 
emphasized.  It  is  better  to  teach  the  system  in  the  seventh 
or  eighth  grade  than  earlier,  because  it  can  be  closely  cor- 
related with  the  work  in  science  and  with  the  industrial 
and  commercial  activities  which  are  usually  emphasized  in 
these  grades. 

Suggestions  on  the  Teaching  of  the  Metric  System 

The  metric  units  in  common  use  should  be  actually  in 
hand.  The  best  results  cannot  be  secured  unless  the  pupils 
see  and  handle  the  measures  and  they  should  be  studied 
as  metric  units,  without  regard  to  their  nearest  English 
equivalents.  Pupils  should  be  trained  to  think  in  terms  of 
metric  units.  The  power  to  estimate  accurately  in  terms 
of  these  units  may  be  developed  through  drill.  Ask  the 
pupils  to  hold  two  fingers  one,  two,  or  three  centimeters 
apart.  Estimate  the  linear  and  surface  measure  of  numer- 
ous common  objects.  Most  of  the  work  in  the  metric 
system  should  be  oral.  After  the  units  are  well  under- 
stood and  some  facility  has  been  acquired  in  their  use,  a 
brief  study  of  the  equivalents  of  the  English  system  may 
be  made.  (A  table  of  equivalents  is  given  at  the  close  of 
this  chapter.) 

The  abbreviations  for  the  metric  units  are  not  uniform 
even  in  France.  Square  meter  is  written  as  qm,  or  m2,  or 
DM.  Each  compound  name  in  the  metric  system  is  accented 
on  the  first  syllable,  thus :  kil'ometer ;  mil'limeter. 

The  prefixes  must  be  thoroughly  learned  if  one  is  to 
master  the  metric  system.  The  pupil  must  know  that 
"kilo"  always  means  1,000,  whether  it  is  prefixed  to  the 


THE  METKIC  SYSTEM  287 

unit,  meter,  liter,  or  gram.  A  kilogram  is  1,000  grams; 
a  kil'ometer  is  1,000  meters;  a  kil'oliter  is  1,000  liters. 
Similarly,  "milli"  when  prefixed  to  any  units  means  .001 
of  that  unit.  Thus,  millimeter  means  .001  of  a  meter.  The 
following  mnemonics  may  help  the  beginner  to  associate 
the  new  terms  with  names  which  are  familiar.  The  word 
" meter"  is  familiar  to  the  pupil  in  such  terms  as  " water 
meter, ' '  "  gas  meter, ' '  and  ' '  cyclometer. ' '  The  term  always 
means  "a  measure."  Deci,  meaning  .1,  may  be  associated 
with  a  dime,  0.1  of  a  dollar.  Centi,  meaning  .01,  may  be 
associated  with  cent,  .01  of  a  dollar.  Milli,  meaning  .001, 
may  be  associated  with  mill,  .001  of  a  dollar.  If  the  term 
dekagram,  a  ten-sided  figure,  is  familiar  to  the  pupil,  he 
will  more  easily  remember  that  deka  means  ten.  Myria  is 
easily  remembered  as  a  large  multiplier  by  associating  it 
with  the  word  myriad. 

The  prefixes  should  be  drilled  upon  until  they  are  mas- 
tered and  the  pupil  is  able  to  translate  back  and  forth  in 
the  table.  The  prefixes  which  indicate  multiples  of  the  unit 
are  derived  from  Greek,  and  those  which  indicate  the 
decimal  divisions  of  the  unit  are  from  Latin. 

The  remarkable  simplicity  of  the  metric  system  and  the 
ease  with  which  reductions  may  be  made  should  be  empha- 
sized. For  example,  compare  the  ease  with  which  487.26 
kilometers  may  be  reduced  to  meters  with  a  similar  reduc- 
tion in  the  English  system.  Reductions  in  the  metric 
system  involve  only  the  moving  of  the  decimal  point ;  thus 
the  subject  gives  an  opportunity  to  review  decimals.  Since 
1,000  liters  equal  1  cubic  meter,  the  capacity  of  bins  and 
tanks  may  be  computed  with  much  greater  ease  than  by 
use  of  the  English  units. 

The  five-cent  piece  may  be  used  to  illustrate  some  of 
the  metric  units.  Its  diameter  is  2  centimeters;  its  thick- 
ness is  2  millimeters,  and  its  weight  is  5  grams, 


288  HOW  TO  TEACH  ARITHMETIC 

The  definitions  of  meter,  liter,  and  gram  should  be 
learned. 

A  meter  is  .0000001  of  the  distance  from  the  equator  to 
the  pole.  It  is  the  unit  of  length  and  is  the  base  of  the 
entire  metric  system. 

A  liter  (pronounced  "leter")  is  the  capacity  of  a  cube  .1 
meter  on  an  edge.  It  is  the  unit  of  capacity. 

A  gram  is  the  weight  of  a  cube  of  distilled  water  .01  of  a 
meter  on  an  edge.  It  is  the  unit  of  weight. 

Prefix      Meaning  Illustration     Meaning 

myria       10,000  myriameter     10,000  meters 


kilo 
hekto 
deka 

1,000 
100 
10 

kilogram 
hektogram 
dekameter 

1,000  grams 
100  grams 
10  meters 

deci 

0.1 

deciliter 

0.1  liter 

centi 

0.01 

centimeter 

0.01  meter 

milli  0.001         millimeter  0.001  meter 

mikro  0.000001  mikrometer  0.000001  meter 

Table  of  Equivalents 

A  meter  =39.37  inches  =  3*4  ft.  nearly 

=  3  ft.  3  inches,  %  inch, 
nearly. 

A  Liter  =  1  quart  nearly 

A  kilogram  =  2.2  pounds  nearly 

A  gram  =  15.43  grains 

A  hektare  =  2.47   acres 

The  meter  is  used  for  dry  goods,  merchandise,  engineer- 
ing, construction,  and  other  purposes  where  the  yard  and 
foot  are  used  in  the  English  system. 

The  centimeter  and  millimeter  are  used  instead  of  the 
inch  and  its  fractions  in  machine  construction  and  similar 
work. 


THE  METRIC  SYSTEM  289 

The  centimeter  is  used  in  expressing  sizes  of  paper  and 
books. 

Any  quantity  consisting  of  several  denominations  may 
be  treated  as  an  integer  and  a  decimal,  the  decimal  point 
separating  the  unit  and  its  divisions.  For  example, 
1420.25  meters  is  not  read  1  kilometer,  4  hektometers,  2 
dekameters,  2  decimeters,  and  5  centimeters,  but  is  read 
1,420  meters  and  25  centimeters,  just  as  $1420.25  is  read 
$1420  and  25  cents. 

Unit  Where  Employed 

Megameter  Astronomy 

Myriameter  Geography 

Kilometer  Distances  in  general 

Hektometer  Artillery 

Dekameter  Surveying 


Decimeter 

Centimeter 

Millimeter 

Mikron 

Millimikron 


Commerce 

Industry  and 

Science 

Metrology 

Spectroscopy 

Microscopy 


CHAPTER  XIX 
INVOLUTION  AND  EVOLUTION 

The  subjects  of  Involution  and  Evolution  were  treated 
in  some  of  the  early  texts,  but  most  of  the  medieval  arith- 
metics did  not  discuss  the  subject  of  involution  extensively. 
The  modern  treatment  of  this  subject  was  introduced  from 
algebra.  Evolution,  as  studied  in  the  grades,  presupposes 
but  little  knowledge  of  involution,  and  the  movement 
towards  the  elimination  of  topics  will  probably  reduce  the 
treatment  of  involution. 

Neither  involution  nor  evolution  is  frequently  used  in 
business.  A  knowledge  of  squares  and  cubes,  of  square 
roots  and  cube  roots,  is  necessary  in  certain  scientific  work, 
but  the  scientist  and  the  engineer  use  tables  or  the  slide 
rule  to  secure  the  results  desired  in  the  computation. 

Cube  root  has  been  eliminated  from  most  courses  in 
arithmetic.  A  recent  report  indicates  that  the  subject  is 
still  taught  in  about  28%  of  the  elementary  schools,  but  it 
will  no  doubt  be  eliminated  from  many  of  these  schools 
within  the  next  few  years.  The  study  of  cube  root  is  being 
postponed  until  the  pupil  encounters  the  subject  in  algebra, 
or  is  omitted  entirely.  Square  root  is  taught  in  most 
schools  in  the  eighth  grade.  The  chief  reason  for  its  reten- 
tion in  the  course  is  the  fact  that  it  is  needed  in  the  mensu- 
ration of  many  geometrical  forms.  For  example,  a  knowl- 
edge of  square  root  is  necessary  in  finding  the  diagonal  of 
of  a  rectangle  or  of  a  cube,  the  altitude  and  area  of  an 
isosceles  triangle,  and  the  slant  height  of  a  pyramid  or 
cone.  If  square  root  is  applied  to  the  solution  of  some 
real  problems,  pupils  will  be  interested  in  it. 

290 


INVOLUTION  AND  EVOLUTION  291 

The  Terms  Used 

The  technical  terms  of  involution  and  of  evolution  should 
be  thoroughly  understood.  The  meaning  of  power,  expo- 
nent, square  and  square  root,  radical  sign  and  digit  should 
be  familiar  to  the  pupil.  Much  time  in  computation  will 
be  saved  if  the  pupil  is  required  to  learn  the  squares  of  all 
integers  from  1  to  20  or  25,  and  the  cubes  from  1  to  10. 
Pupils  should  be  able  to  state  at  sight  the  square  roots  of 
such  numbers  as  the  following : 

64        400        900        81        36        3600 

Square  Boot  ly  Factoring 

The  first  examples  worked  in  square  root  should  be  by 
factoring.  The  fact  should  be  emphasized  that  we  seek  to 
find  two  equal  numbers  whose  product  is  the  given  number. 
2025  =  5x5x9x9,  or  52  x  92 ;'  therefore  the  square  root  of 
2025  =  5x9  =  45.  Similarly,  324  =  2x2x9x9,  or  22x92; 
therefore  the  square  root  of  324  =  2x9  =  18.  Similarly, 
53361  =  3x3x7x7x11x11,  or  32x72xll2;  therefore  the 
square  root  of  53361  =  3x7x11  =  231. 

Sometimes  we  wish  to  obtain  the  square  root  of  a  number 
not  readily  factored,  or  the  approximate  square  root  of  a 
number  that  is  not  a  perfect  square.  If  we  wished  to  find 
the  square  root  of  5329,  or  the  approximate  square  root  of  3, 
the  desired  result  could  not  be  conveniently  found  by  fac- 
toring. 

Other  Methods  for  Extracting  Square  Root 

There  are  two  general  methods  for  explaining  the  extrac- 
tion of  the  square  root  of  such  numbers.  One  of  these  is 
known  as  the  algebraic  and  the  other  as  the  geometric 
method.  One  of  the  methods  is  analytic  and  the  other  is 
synthetic.  Text-book  writers  are  not  agreed  as  to  which 
method  should  be  used  in  explaining  the  process  of  evolu- 


292  HOW  TO  TEACH  ARITHMETIC 

tion,  and  several  of  the  best  books  give  both  methods. 
Occasionally  a  teacher  advocates  the  teaching  of  square 
root  as  a  purely  mechanical  process,  with  no  attempt  to 
justify  the  procedure  to  the  mind  of  the  pupil.  Such  a 
point  of  view  is  contrary  to  the  best  educational  theories 
of  the  day.  It  is  not  assumed  that  the  explanation  of 
square  root  by  either  the  analytic  or  the  synthetic  method 
will  so  impress  itself  upon  the  mind  of  every  pupil  that 
each  one  will  be  able  to  explain  the  theory  involved  in  a 
given  problem.  Pupils  should  understand  each  step  of  the 
process,  but  the  object  is  not  to  enable  the  pupil  to  repeat 
the  explanation  in  a  more  or  less  mechanical  way  ;  it  is  to 
justify  the  various  steps  to  his  mind.  After  this  has  been 
done  the  pupil  should  be  required  to  work  numerous  exam- 
ples until  he  has  acquired  facility  in  the  process.  Pupils 
should  be  required  to  formulate  a  rule  for  square  root  and 
to  memorize  it. 

Algebraic  Method.  This  method  is  applicable  to  the 
extraction  of  any  desired  root,  but  no  root  higher  than  the 
third  is  necessary  in  arithmetic. 

The  square  of  45  may  be  found  as  follows: 

(45)2  =  (40  +  5)2  Similarly,  (97)2=  (90  +  7)2 
40+5  90+7 

40+5  90  +7 

402+     (40x5)  902+     (90x7) 

+     (40x5)+52  +     (90x7)+72 


402  +  2  (40  x  5)  +  52  902  +  2  (90  x  7)  +  T2 

Every  number  composed  of  two  or  more  digits  may  be 
regarded  as  composed  of  tens  and  units.  Thus,  45  equals 
4  tens  plus  5  units;  97  equals  9  tens  plus  7  units.  434 
equals  43  tens  plus  4  units.  By  working  several  examples 
similar  to  the  above,  we  find  that  the  square  of  any  number 
will  equal  the  square  of  the  tens,  plus  twice  the  tens  by 


INVOLUTION  AND  EVOLUTION  293 

the  units,  plus  the  square  of  the  units.  If  t  represents  the 
number  of  tens  in  any  given  number  and  u  represents  the 
number  of  units,  we  may  say  that  the  square  of  the  number 
equals  t2  +  2tu  +  u2. 

Since  the  square  of  the  smallest  number  of  one  digit  con- 
tains one  digit,  and  the  square  of  the  largest  number  of 
one  digit  contains  two  digits,  the  square  of  any  number 
of  one  digit  must  contain  either  one  or  two  digits.  Simi- 
larly, since  the  square  of  10,  the  smallest  number  of  two 
digits,  is  100,  and  the  square  of  99,  the  largest  number  of 
two  digits,  is  9801,  we  conclude  that  the  square  of  a  number 
of  two  digits  must  contain  three  or  four  digits.  This  may 
be  extended  to  larger  numbers.  If  any  integral  number  is 
divided  into  groups  of  two  digits  each,  from  the  right  to 
the  left,  the  number  of  digits  in  the  root  will  be  the  same 
as  the  number  of  groups  of  digits.  The  last  group  to  the 
left  may  contain  one  or  two  digits.  Thus  the  square  root 
of  18  14  76  contains  three  digits.  The  square  root  of  169 
contains  two  digits. 

Required  to  find  the  square  root  of  2025. 

If  2025  is  a  perfect  square,  its  square  root 
_A  A  must  contain  two  digits,  since  the  number  itself 
contains  four  digits.  The  largest  square  in  20 
is  16,  and  the  square  root  of  16  is  4.  The  ten's 
digit  of  the  root  is  therefore  4.  The  square  of 
4  tens,  or  40,  is  1600.  425  is  composed  of  two 


times  the  tens,  times  the  units,  plus  the  square 
oi  the  units.  We  do  not  know  the  square  of  the  units,  but 
we  know  that  it  is  small  compared  with  two  times  the  tens 
times  the  units.  We  may  therefore  think  of  425  as  com- 
posed approximately  of  two  factors;  one  of  these  factors 
(two  times  tens)  is  known,  and  the  other  is  not  known.  We 
therefore  divide  425  by  two  times  tens  in  order  to  find  the 
units,  which  is  the  other  factor.  In  this  division  we  remem- 


294  HOW  T0  TEACH  ARITHMETIC 

her  that  425  is  somewhat  larger  than  the  product  of  these 
factors,  hence  we  make  a  slight  •  allowance  for  this  excess. 
In  the  problem  above,  two  times  the  tens  equals  2x40,  or 
80.  80  is  contained  in  425  five  times.  The  units  digit  of 
the  root  is  5.  The  5  is  added  to  the  80  and  this  result  is 
then  multiplied  by  5,  because  we  may  regard  two  times  the 
tens  times  the  units  plus  the  square  of  the  units  as  (two 
times  the  ten  plus  the  units)  times  units. 

In  practice  we  may  omit  the  zeroes  in  the  square  of  40, 
also  the  zero  of  80.  We  may  then  annex  the  5  to  the  8,  and 
the  work  appears  as  follows  : 


_ 

2025 
16 


85 


425 
425 


The  method  just  explained  may  be  used  in  finding  the 
square  root  of  any  perfect  square  or  the  approximate  square 
root  of  a  number  whose  exact  square  root  cannot  be  found. 
If  the  given  number  contains  more  than  two  groups  of  two 
digits  each,  we  may  think  of  the  part  of  the  root  found  as  so 
many  tens  with  reference  to  the  next  digit  to  be  found,  and 
the  process  is  the  same  as  before.  This  should  be  clear 
from  the  illustration  that  follows : 

Required  to  find  the  square  root  of  107584. 

_3_2_8 
10  75  84 
9 


175 
124 


648 


5184 
5184 

The  square  root  of  107584  is  therefore  328. 


INVOLUTION  AND  EVOLUTION  295 

The  rule  for  grouping  the  digits  of  a  decimal  whose 
square  root  is  required  may  be  easily  developed  by  a  method 
similar  to  the  one  followed  in  determining  how  to  group  the 
digits  of  an  integral  number.  Since  the  square  of  tenths  is 
hundredths,  and  the  square  of  hundredths  is  ten  thou- 
sandths, we  conclude  that  in  extracting  the  root  of  a  num- 
ber containing  a  decimal  fraction  we  must  begin  at  the 
decimal  point  and  group  the  digits  in  twos  from  the  left 
to  the  right.  Thus,  .57  46  28;  a  zero  may  be  annexed 
to  complete  the  number  of  digits  in  the  last  group. 
Thus,  .38  75"  2  =  .38"  75  20.  With  the  exception  of  this 
difference  in  grouping  the  digits,  the  extraction  of  square 
root  is  practically  the  same  for  decimal  fractions  as  for 
integers.  In  extracting  the  square  root  of  decimal  frac- 
tions, the  chief  difficulty  is  encountered  in  such  examples 
as  the  following:  \A3;  \A124;  V^5-  The  difficulty  here 
is  usually  due  to  a  mistake  in  pointing  off,  or  grouping  the 
digits.  If  a  cipher  is  annexed,  so  that  the  number  con- 
tains an  even  number  of  decimal  places,  this  difficulty  is 
removed.  _  For  example,  V^3  =  V^O;  V^i^  V-1240; 


Not  all  numbers  are  perfect  squares,  hence  not  all  num- 
bers can  be  expressed  as  the  product  of  two  equal  factors. 
For  example,  if  we  wish  to  find  the  square  root  of  2,  we 
may  annex  zeroes  to  the  right  of  the  decimal  point  and 
carry  the  extraction  of  the  root  to  any  required  degree  of 
accuracy.  V~2  =  1.414  ;  V3  =  1.732.  The  square  roots  of  2 
and  of  3  are  so  frequently  used  in  mensuration  that  they 
should  be  memorized. 

If  we  define  the  square  root  of  an  expression  to  be  one 
of  its  two  equal  factors  it  is  evident  that  only  abstract 
numbers  can  have  square  roots.  No  number  multiplied 
by  itself  equals  9  square  feet,  or  $9,  or  9  books.  Hence 
9  square  feet,  $9,  and  9  books  have  no  square  roots. 


D 

M 

40                Fs  ( 

40 

R 

40 

A 

H 

296  HOW  TO  TEACH  ARITHMETIC 

Geometric  Method 

The  extraction  of  the  square  root  of  2025  may  be  ex- 
plained by  the  use  of  a  diagram,  and  this  method  of 
explanation  may  be  used  in  any  problem  in  square  root. 

Suppose  that  ABCD  is 
a  square  containing  2025 
square  units.  It  is  re- 
quired to  find  the  length 
of  one  side  of  this  square. 
A  square  whose  area  is 
1600  square  units  is  40 
units  on  each  side,  and  a 
square  whose  area  is  2500 
square  units  is  50  units  on 
each  side.  The  side  of  the 
square  ABCD  must  there- 
fore contain  between  40  and  50  units.  If  a  square  whose 
side  is  40  units  is  taken  out,  then  1600  square  units  are 
taken  out  and  the  remaining  area  is  425  square  units. 
In  the  figure,  AHEM  is  40  units  on  each  side,  and  there- 
fore contains  1600  square  units.  Since  the  square  ABCD 
contains  2025  square  units,  the  rectangle.  MRFD  plus 
rectangle  RHBE  plus  the  square  FREC  contains  425  square 
inches.  The  combined  length  of  the  two  rectangles  is 
known  to  be  80  units.  The  problem  is  to  find  how  wide 
these  rectangles  and  the  square  must  be  so  that  their  com- 
bined area  shall  be  425  square  units.  If  the  width  were 
6  units,  the  combined  area  would  be  more  than  6x80 
square  units,  or  480  square  units.  Try  5  units.  The  com- 
bined area  of  the  rectangles  would  then  be  5x80  square 
units,  or  400  square  units,  and  the  area  of  the  square  would 
be  5x5  square  units,  or  25  square  units.  The  combined 
area  of  the  rectangles  and  the  square  is,  therefore,  425 


INVOLUTION  AND  EVOLUTION  297 

square  units.  This  is  known  to  be  the  area  of  the  remainder 
of  the  square  ABCD  after  the  area  of  the  square  AHRM 
has  been  considered.  The  length  of  AB  is  therefore  40 
units  plus  5  units,  or  45  units. 

It  should  be  noticed  in  the  above  explanation  that  the 
number  of  units  in  the  combined  areas  of  the  two  rect- 
angles and  the  square  is  the  remainder  after  the  number 
of  units  in  the  square  AHRM  has  been  subtracted.  The 
sum  of  the  number  of  units  in  the  length  of  the  two 
rectangles  is  the  trial  divisor,  and  the  number  of  units  in 
the  side  of  the  square  RECF  is  the  complete  divisor. 

The  square  root  of  any  perfect  square  or  the  approxi- 
mate square  root  of  any  number  not  a  perfect  square  may 
be  found  by  this  method.  Whether  square  root  is  taught 
by  the  algebraic  or  by  the  geometric  method,  the  pupils 
should  be  asked  to  formulate  a  statement  of  the  necessary 
steps  in  the  process. 

Such  a  statement  as  the  following  is  valuable  as  a  work- 
ing rule:  1.  Separate  the  number  into  periods  of  two 
digits  each,  beginning  at  the  decimal  point.  2.  Find  the 
greatest  square  in  the  left-hand  period  and  subtract  it, 
bringing  down  the  next  period.  3.  Divide  the  remainder 
by  twice  the  part  already  found.  4.  To  this  divisor  add 
the  number  thus  found  and  multiply  this  sum  by  the  num- 
ber found.  5.  Subtract  this  result  and  bring  down  the  next 
period.  Proceed  as  in  steps  3  and  4. 

Square  Root  of  Fractions 

The  square  root  of  a  common  fraction  both  of  whose 
terms  are  perfect  squares  may  best  be  found  by  extracting 
the  square  root  of  both  terms.  Thus : 


298  HOW  TO  TEACH  ARITHMETIC 

If  the  terms  are  not  perfect  squares,  the  square  root  may 
be  found  in  two  ways:  (a)  by  reducing  the  fraction  to  a 
decimal  and  then  extracting  the  root  to  the  required  num- 
ber of  decimal  places,  or  (b)  by  multiplying  both  terms  of 
the  fraction  by  a  number  that  will  make  the  denominator 
a  perfect  square,  and  then  proceeding  as  in  the  illustrations 
of  method  "B"  below. 

Illustrations  of  Method  "A" 
=  .4  VA=  .632 


f  =  .428571;  V.428571  -  .654 
Illustrations  of  Method  "B" 


7 

Problems  involving  the  applications  of  square  root  occur 
in  the  subject  of  Mensuration. 


CHAPTER  XX 

RATIO   AND   PROPORTION 

One  number  may  be  compared  with  another  in  two 
ways ;  we  may  inquire  how  much  greater  or  less  one  num- 
ber is  than  another,  or  we  may  inquire  how  many  times 
one  number  equals  another. 

The  ratio  of  two  similar  quantities  is  the  measure  of  the 
relation  of  the  quantities.  The  ratio  of  6  to  3  is  2.  The 
first  term  of  a  ratio  is  called  the  antecedent;  the  second 
term  is  called  the  consequent.  The  ratio  expresses  how 
many  times  the  consequent  must  be  taken  to  produce  the 
antecedent.  The  ratio  of  6  to  3  is  2  because  3  must  be 
taken  twice  to  produce  6. 

Proportion  arises  from  a  comparison  of  ratios;  it  is, 
therefore,  based  upon  comparison.  "It  is  a  comparison  of 
the  results  of  two  previous  comparisons/'  Each  ratio 
involves  a  comparison,  and  the  statement  that  the  two 
ratios  are  equal  involves  a  third  comparison. 

Since  a  proportion  is  the  expression  of  equality  of  two 
equal  ratios,  it  is  an  equation. 

Proportions  were  formerly  written  by  use  of  the  sym- 
bol ::.  Thus 

4:6::2:3 

The  symbol  : :  is  rapidly  disappearing  and  the  sign  of 
equality  is  being  used  instead.  Thus 

(a)  4:6  =  2:3 

(b)  *  =  f 

The  equational  form  (b)  is  now  the  one  in  most  common 
use. 

299 


300  HOW  TO  TEACH  AEITHMETIC 

It  is  easier  in  practice  to  place  the  unknown  quantity  in 
the  first  term. 

"When  proportion  is  used  it  should  be  as  a  reasoning  proc- 
ess and  not  as  a  mere  mechanical  procedure  to  secure  an 
answer. 

The  fundamental  principle  involved  in  proportion  is  that 
the  product  of  the  means  is  equal  to  the  product  of  the 
extremes.  Because  of  this  principle  if  any  three  terms  of 
a  proportion  are  known  the  fourth  may  be  found.  Pro- 
portion was  formerly  called  "The  Eule  of  Three,"  and 
was  regarded  as  one  of  the  most  important  parts  in 
arithmetic. 

Humpfrey  Baker,  1562,  speaking  of  proportion  said, 
* '  The  rule  of  three  is  the  chief est  and  most  profitable  and 
the  most  excellent  rule  of  all  arithmetike,  for  which  cause 
it  is  said  philosophers  did  name  it  the  golden  rule." 

The  increasing  use  of  the  simple  equation  and  of  analy- 
sis is  relegating  the  subject  of  ratio  and  proportion  to  a 
position  of  subordinate  importance. 

The  present  tendency  to  eliminate  from  the  text  in 
arithmetic  those  problems  that  do  not  in  some  way  relate 
to  real  life  is  causing  most  of  the  old  problems  in  propor- 
tion to  be  omitted. 

The  problem  which  states  that  14  men  can  dig  a  ditch 
8  feet  wide,  7  feet  deep  and  600  feet  long  in  22  days, 
working  8  hours  a  day,  and  requires  the  pupil  to  find  out 
how  many  days  it  would  require  9  men  to  dig  a  ditch  10 
feet  wide,  8  feet  deep  and  480  feet  long,  working  7  hours 
a  day,  is  omitted  from  most  texts  to-day. 

Proportion  can  be  used  to  advantage  in  some  of  the 
applications  of  arithmetic,  especially  in  the  solution  of 
problems  relating  to  similar  figures. 


CHAPTER  XXI 
MENSURATION 

Why  Mensuration  is  Taught 

An  examination  of  the  literature  treating  mensuration 
shows  that  the  presence  of  the  subject  in  the  elementary 
schools  is  justified  by  widely  divergent  points  of  view. 
Running  through  the  arguments  advanced  by  some  of  the* 
most  distinguished  mathematicians  is  the  assumption  that 
the  mensuration  of  the  elementary  schools  should  be  re- 
garded as  a  segment  of  a  great  system  of  thought.  Nat- 
urally, therefore,  they  urge  that  it  be  taught  as  intro- 
ductory and  preparatory  to  geometry.  Emphasized  thus, 
the  purpose  of  mensuration  is  to  give  insight  into  higher 
mathematical  relations. 

Opposed  to  this  point  of  view  is  that  of  some  of  the 
more  extreme  theorists  of  present  day  education,  who  hold 
that  every  subject  can  be  justified  in  the  curriculum  only 
on  the  ground  of  its  immediate  utilitarian  value.  If  in- 
timacy between  life  and  the  text  instead  of  the  exposition 
of 'a  great  system  of  deductive  reasoning  is  the  criterion 
of  the  presence  of  material  in  the  curriculum,  then  much 
that  has  commonly  appeared  under  the  title  of  mensuration 
in  our  books  should  be  eliminated. 

The  wise  school  administrator  and  well  informed  teacher, 
will  see  that  both  aspects  of  the  subject  receive  attention, 
but  that  neither  is  overemphasized.  The  great  mathema- 
tician may  be  absorbed  by  the  beautiful  system  of  logical 
reasoning  found  in  geometry,  but  that  is  not  excuse  enough 
for  teaching  mensuration  as  an  exemplification  of  the  sys- 

301 


302  HOW  TO  TEACH  AEITHMETIC 

tern.  A  teacher  may  attempt  to  articulate  the  facts  and 
theories  of  the  school  with  community  activities,  as  indeed 
she  should;  but  that  gives  her  no  warrant  for  neglecting 
the  preparatory  character  of  mensuration. 

Mensuration  affords  every  teacher  a  great  opportunity 
of  opening  the  way  to  new  and  inviting  fields.  It  is  just- 
tified  partly  because  it  does  point  the  way  to  a  field  of 
more  elaborate  constructive  thought,  partly  because  it  is 
immediately  useful,  but  more  especially  because  a  knowl- 
edge of  its  materials  aids  one  to  properly  interpret  the 
natural  features  of  the  world.  Instruction  in  it  should 
aid  in  securing  an  organic  conception  of  certain  physi- 
cal features  of  the  world  in  which  we  live.  Dominated  by 
such  an  aim  it  would  be  impossible  to  teach  the  subject 
without  supplying  an  abundance  of  valuable  information 
about  computing  the  contents  of  solids  or  the  areas 
of  surfaces.  It  is  through  a  mastery  of  such  facts  that 
there  gradually  dawns  upon  the  pupil  a  knowledge  of  the 
integral  character  of  the  natural  world.  Mensuration  is 
not  taught  primarily  to  give  facts  and  skill  unrelated  to 
life,  but  to  assist  pupils  to  interpret  rationally  the  universe 
about  them. 

Method  to  be  Employed 

This  raises  the  important  question  as  to  the  method  that 
shall  be  employed  in  teaching  the  subject.  Shall  its  truths 
be  demonstrated,  illustrated,  or  taught  by  rote  or  formula? 
Throughout  this  entire  book  we  have  argued  against  teach- 
ing by  formula.  We  believe,  however,  that  mensuration 
suffers  more  than  any  other  topic  in  arithmetic  in  this 
particular,  for  the  reason  that  a  large  percentage  of  the 
teaching  population  have  no  adequate  notion  of  its  signifi- 
cance. When  teachers  as  a  class  are  better  qualified  aca- 
demically and  professionally  for  the  work  they  profess  to 


MENSURATION  303 

do,  mensuration  will  be  taught  on  a  higher  conscious  level 
than  now.  It  will  no  longer  be  a  habit  subject  to  be 
acquired  through  drill,  but  a  thought  subject  to  be 
developed. 

Certainly  no  comprehensive  notion  of  the  interpretative 
value  of  mensuration  is  possible  unless  its  truths  are  both 
illustrated  and  demonstrated.  By  the  time  the  subject  is  in- 
troduced in  the  seventh  or  eighth  grade,  the  children  already 
have  a  fairly  liberal  notion  of  its  elementary  phases.  They 
learned  in  the  lower  grades  that  mensuration  deals  with 
lines,  surfaces,  and  volumes.  It  remains  to  apply  the 
ideas  thus  acquired  in  the  earlier  grades  to  objects  of 
greater  utility,  such  as  fields,  cisterns,  cellars,1  and  the  like. 
It  is  very  necessary  that  the  measurements  involved  in 
these  larger  figures  be  clearly  understood;  otherwise  both 
pupils  and  teachers  will  continue  to  make  the  wildest 
guesses  about  distances,  areas  and  contents. 

Measurement  of  Plane  Figures 

The  fact  that  of  all  plane  figures  the  rectangle  is  the 
only  one  whose  area  may  be  directly  found  by  applying 
the  unit  of  measure  (a  square)  and  then  counting  the  num- 
ber of  times  it  is  applied,  should  be  developed  and  em- 
phasized. The  rectangle  is  the  plane  figure  from  which 
the  mensuration  of  all  other  polygons  is  developed.  The 
area  of  rectangles,  having  been  studied  in  the  earlier 
grades,  should  be  profitably  reviewed  at  this  point  as  a  basis 
for  the  development  of  the  mensuration  of  other  polygons. 
Attention  should  be  directed  to  the  fact  that  a  square  one 
unit  long  and  one  unit  wide  contains  one  square  unit,  and 
that  a  rectangle  two  units  long  and  one  wide  contains  two 
square  units  or  two  times  one  square  unit,  which  is  two 
square  units.  Many  teachers  still  permit  their  pupils  to 
say  the  area  of  a  rectangle  2  inches  long  and  1  inch  wide 


304 


HOW  TO  TEACH  AEITHMETIC 


is  2  inches  x  1  inch  =  2  square  inches.  The  area  of  a  rectan- 
gle whose  length  is  4  feet  and  whose  altitude  is  3  feet  should 
be  expressed  as  4x3x1  sq.  ft.,  or  as  4x3  sq.  ft.  Such  a 
statement  gives  the  correct  result,  and  does  not  violate  any 
of  the  principles  of  multiplication. 

Enough  class  exercises  should  be  given  to  insure  a  clear 
notion  of  area.  Pupils  may  be  required  to  estimate  the 
number  of  square  inches  in  their  book  cover,  a  window, 
a  picture,  a  blackboard,  and  then  to  test  the  accuracy  of 
each  estimate.  The  ability  to  estimate  area  in  square  inches, 
square  feet  and  square  yards  may  be  trained  by  cutting, 
drawing  and  by  estimating  and  verifying;  of  square  rods 
and  acres  by  actual  measurement;  of  a  square  mile  or 
section  of  land,  by  viewing  it  or  walking  around  it.  Later 
actual  practical  problems  involving  area  should  be  given. 
Perhaps  the  most  practical  of  these  deal  with  land  measure 
and  surveying,  flooring,  carpentering,  lathing,  plastering 
and  papering. 

The  Parallelogram 

After  the  pupil  is  familiar  with  the  mensuration  of  the 
rectangle  he  should  be  taught  that  the  area  of  any 
parallelogram  is 
e  q  u  i  v  a  lent  to 
that  of  a  rect- 
angle of  the 
same  base  and 
altitude  as  the 
p  a  r  a  llelogram. 
This  may  be  de- 
duced by  a  com- 


M 


B 


parison    of    the 

figures.     If  we  draw  a  perpendicular  from  C  to  AB  and 

then  cut  out  the  triangle  CMB  and  put  it  in  the  position  of 


MENSURATION  305 

the  triangle  ADH,  we  shall  have  the  rectangle  DHMC. 
This  rectangle  is  evidently  composed  of  the  same  parts  as 
the  original  parallelogram  ABCD,  and  is,  therefore,  equal 
to  it  in  area.  Since  the  rectangle  and  parallelogram  have 
bases  of  equal  length  and  also  have  equal  altitudes  the  area 
of  the  parallelogram  may  be  stated  in  terms  of  the  rectan- 
gle. It  should  be  noted,  however,  that  the  area  of  a  gen- 
eral parallelogram  cannot  be  found  by  applying  the  square 
unit  of  measure,  because  the  square  unit  will  not  exactly 
fit  at  the  vertices.  At  this  stage  of  the  work,  the  pupil  can 
only  infer  that  the  figure  is  the  equivalent  of  a  rectangle 
because  it  appears  to  be  so.  A  rough  proof  of  the  equiv- 
alence of  an  oblique  parallelogram  and  rectangle  can  be 
secured  by  folding  and  cutting  off  one  right  triangle  and 
transposing  it  to  the  opposite  end  of  the  parallelogram. 
On  the  other  hand  the  teacher  should  not  overlook  the 
opportunity  to  tell  the  pupil  that  when  he  learns  geometry 
he  will  be  able  to  prove  conclusively  that  HMCD  is  a 
rectangle. 

Measurement  of  Triangles 

The  manner  in  which  the  various  kinds  of  triangles  are 
related  in  area  to  parallelograms  and  rectangles,  should 
be  developed.  Before  this  is  done,  perhaps,  the  attention 
of  pupils  should  be  called  to  the  classification  of  triangles 
on  the  basis  of  the  comparative  length  of  their  sides  or  the 
size  of  the  angles.  The  essential  differences  in  triangles 
may  be  discovered  by  the  children  themselves,  but  the 
names  of  the  various  classes  or  kinds  of  triangles  should 
be  told  the  children.  An  equilateral  triangle  has  all  of  its 
three  sides  equal.  An  isosceles  triangle  has  two  of  its  sides 
equal.  A  scalene  triangle  has  no  two  sides  equal.  Triangles 
are  also  called  acute-angled,  obtuse-angled,  or  right-angled, 


306 


HOW  TO  TEACH  AEITHMETIC 


the  classification  depending  upon  the  size  of  the  largest 
angle. 

In  order  to  emphasize  and  fix  these  conceptions  such 
questions  as  the  following  may  be  asked:  Is  an  isosceles 
triangle  necessarily  equilateral?  May  an  isosceles  triangle 
be  equilateral  ?  "Which  is  the  more  general  term,  equilateral 
or  isosceles  ?  May  a  right  triangle  be  isosceles  ?  May  it  be 
equilateral?  May  an  obtuse-angled  triangle  be  equilateral 
or  isosceles? 

The  simplest  case  showing  the  relation  of  a  triangle  to 
the  rectangle  is  the  half  square.  It  seems  evident  to  a 
child  that  a  diagonal  divides  the  square  into  halves.  This 
fact  can  be  illustrated  by  the  folding  of  paper. 

The  equilateral  triangle  ABC  may  be  cut  up  as  indicated 
in  the  figure,  the  triangle  CDQ  being  put  in  the  position 
ARQ  and  the  triangle 
CDX  in  the  position  XMB 
and  the  result  is  the  rect- 
angle ABMR. 

It  can  be  also  shown  that 
an  equilateral  triangle  has 
half  the  area  of  a  parallelo- 
gram having  the  same  base 
and  altitude  as  the  tri- 
angle. By  numerous  illus- 
trations similar  to  these  the  principle  for  finding  the  area 
of  a  triangle  may  be  developed. 


MENSURATION  307 

Measurement  of  Trapezoid  and  Trapezium 

The  method  of  finding  the  area  of  a  trapezoid  may  be 
developed  in  either  of  two  ways :  The  points  0  and  E  are 
the  midpoints  on  the  lines  DA  and  CB  between  the  par- 
allels DC  and  AB. 

By  cutting  off  the  triangle  BHE  and  placing  it  in  the 
position  CME,  and  placing  the  triangle  AOK  in  the  posi- 


tion ROD,  the  result  is  the  rectangle  KHMR.  The  alti- 
tude of  the  trapezoid  has  not  been  altered  and  the  sum  of 
the  bases  of  the  original  trapezoid  ABCD  is  equal  to  the 
sum  of  the  bases  of  the  resulting  rectangle. 


T 

> 
/ 

A                       \                    B 

/ 

\ 

/ 

\ 

/ 

The  same  result  may  be  derived  by  constructing  the 
trapezoid  B  equal  to  the  trapezoid  A.  The  result  is  a  par- 
allelogram. The  altitude  of  the  parallelogram  is  the  same 
as  that  of  the  trapezoid,  and  half  of  the  sum  of  the  bases 
of  the  original  trapezoid  is  equal  to  one'  base  of  the  par- 
allelogram. The  area  of  the  trapezoid  can  now  be  stated, 
since  the  area  of  the  parallelogram  is  known. 


308  HOW  TO  TEACH  AEITHMETIC 

Measurement  of  Circles 

Pupils  should  be  required  to  learn  three  formulas  for 
finding  the  area  of  a  circle  after  the  method  commonly 
found  in  most  good  texts  has  somewhat  justified  the  process 
to  the  pupil's  mind.  Since  the  number  of  units  in  the  area 
of  a  circle  is  equal  to  one-half  the  number  of  units  in  the 
circumference  times  the  number  of  units  in  the  radius,  we 

C  r 

have  the  formula  a-— ^— ,  where  a,  c,  and  r  represent  the 

number  of  units  in  area,  circumference,  and  radius  respec- 
tively. From  this  formula  the  other  two  formulas  for  the 
area  of  the  circle  may  be  easily  deduced.  If  the  teacher 
will  have  her  pupils  carefully  measure  the  circumferences 
and  diameters  of  several  circles  of  different  sizes  (a  dollar, 
tin  cup,  or  any  circular  object  whose  dimensions  can  readily 
be  found  will  serve  the  purpose),  and  then  have  the  pupils 
divide  the  length  of  the  circumference  by  that  of  the  diam- 
eter, the  quotient  in  every  case  will  be  found  to  be  about  3^. 
This  ratio  of  the  length  of  the  circumference  to  that  of  the 
diameter  is  proved  in  geometry  to  be  always  the  same,  and 
to  be  approximately  equal  to  3.14159.  For  all  practical 
school  purposes  the  values  3.1416  or  -\2-  are  sufficiently 
accurate. 

Since  the  ratio  of  the  length  of  the  circumference  to  that 
of  the  diameter  is  constant,  a  special  symbol  is  used  to  rep- 
resent this  value.  This  symbol  is  the  Greek  letter  pi.  We 

C  C 

therefore  say  77-=^  °r  7j—  =  7r,  since  D  =  2r,  or  C  =  2irr.    If 

Cr 

we  substitute  for  C  its  value  2-n-r  in  the  formula  a  =  —  we 

shall  have  a=    7rrXr,  or  irr2.    Since  d  =  2r,  therefore  r  = 


MENSURATION  309 

d2 
and  r2  =  -r-  .    If  we  put  this  value  for  r2  in  the  formula  ?rr3, 

we.  shall  have  a  =  TT  —  . 

o  i4ifi/i2 
Since  TT  =  3.1416,  we  have  a^       ^       ,  or  .7854d2. 

The  pupil  should  know  these  three  formulas  for  area: 

s 

C  r,  TT  r2  and  w  d2 


Measurement  of  Rectangular  Solids 

Just  as  the  area  of  most  plane  figures  may  be  found  by 
comparing  them  with  the  area  of  a  rectangle,  so  we  may 
find  the  volume  of  regular  solids  by  comparing  them  directly 
or  indirectly  with  the  volume  of  the  rectangular  solid.  The 
rectangle  is  the  only  plane  figure  whose  area  may  be  deter- 
mined by  applying  the  unit  of  measure,  a  square,  and  then 
counting  the  number  of  times  it  is  contained  in  the  rectan- 
gle whose  area  is  desired,  and  the  rectangular  solid  is  the 
only  solid  whose  volume  can  be  found  by  applying  the 
unit  of  measure,  a  cube,  and  counting  the  number  of  times 
it  is  contained  in  the  solid  whose  volume  is  required. 

If  the  teacher  has  a  few  small  cubes  and  will  build  up 
before  the  class,  or  have  some  pupil  build  up,  rectangular 
solids  of  various  dimensions  the  fact  that  the  number 
of  units  in  the  volume  of  a  rectangular  solid  is  equal  to 
the  product  of  the  number  of  units  in  its  three  dimensions 
can  be  easily  deduced.  Thus,  the  volume  of  a  rectangular 
solid  whose  dimensions  are  4  in.  by  3  in.  by  2  in.  is  easily 
seen  to  be  4  x  3  x  2  x  1  cu.  in.  =  24  cu.  in.  It  is  very  desirable 
that  the  teacher  should  not  permit  the  statement  so  com- 
monly used  in  such  problems  (4  in.  x3  in.  x2  in.  =  24  cu. 
inches).  (See  chapter  on  Accuracy.) 


310  HOW  TO  TEACH  AKITHMETIC 

The  Use  of  Models 

Space  does  not  permit  an  extended  discussion  of  the 
various  methods  whereby  the  volumes  of  the  various  solids, 
commonly  considered  in  the  arithmetic  of  the  grades,  may 
be  shown  to  depend  directly  or  indirectly  upon  the  volume 
of  the  rectangular  solid.  Most  arithmetics  contain  a  suffi- 
ciently detailed  discussion  of  these  facts.  The  best  inter- 
ests of  the  class  will  frequently  be  served  if  the  pupils 
have  access  to  models  of  the  various  solids  that  are  to  be 
studied.  Inexpensive  models  for  this  purpose  can  usually 
be  purchased  at  a  comparatively  small  expense.  Many  of 
the  more  common  solids  can  be  easily  made  out  of  wood  or 
pasteboard  by  the  teacher  or  the  pupils.  In  studying  the 
pyramid,  the  cylinder,  the  cone,  and  the  sphere  it  is  very 
desirable  to  have  models  at  hand,  as  they  will  frequently 
make  clear  some  difficulty  due  to  the  inability  of  the  pupil 
to  properly  image  the  solid  under  consideration.  It  is  un- 
doubtedly true  that  such  models  may  be  used  to  excess 
and  thus  defeat  the  very  purpose  that  they  are  to  serve, 
i.  e.,  to  enable  the  pupil  to  image  clearly  the  form  that  is 
under  consideration.  That  models  may  be  used  to  excess 
in  the  study  of  mensuration  should  not  condemn  the  use 
of  models,  but  should  argue  for  more  wisdom  in  their  use. 
It  is  wise  to  dispense  with  the  use  of  objects  in  mensura- 
tion just  as  soon  as  the  pupil  can  properly  image 
the  figure  under  discussion.  Some  pupils  can  image  much 
more  readily  and  more  distinctly  than  others,  and  the 
teacher  should  be  careful  to  use  the  objective  material  only 
in  those  cases  where  it  seems  imperative. 

.Measurement  of  Cylinder 

It  is  desirable  in  studying  the  lateral  area  of  a  cylinder 
to  have  the  pupil  think  of  a  piece  of  paper  so  cut  that 


MENSURATION  311 

it  will  just  cover  the  lateral  surface  when  wrapped  about 
it.  When  the  paper  is  removed  the  pupil  will  readily  see 
that  it  is  in  the  form  of  a  rectangle.  The  base  of  the 
rectangle  is  equal  in  length  to  the  circumference  of  the 
cylinder  and  the  altitude  of  the  rectangle  is  equal  to  the 
altitude  of  the  cylinder.  Since  the  number  of  units  in  the 
area  of  the  rectangle  can  be  found,  the  number  of  units 
in  the  lateral  surface  of  the  cylinder  can  readily  be  deter- 
mined. Let  the  pupil  try  to  image  the  shape  of  the  paper 
that  could  be  made  to  just  cover  the  lateral  surface  of  a 
right  cone.  He  should  see  that  the  paper  is  in  the  form 
of  a  sector  of  a  circle.  From  this  the  formula  for  the 
lateral  surface  of  a  cone  may  be  determined. 

Measurement  of  Pyramids  and  Prisms 

For  comparing  the  volume  or  cubical  contents  of  a  right 
pyramid  with  that  of  a  right  prism  of  equal  base  and 
altitude  it  is  well  to  fill  the  pyramid  with  some  substance 
such  as  sand  and  pour  it  into  the  prism.  When  the  pupil 
finds  that  the  pyramid  must  be  filled  three  times  and  the 
contents  poured  into  the  prism  in  order  that  the  latter 
may  be  full,  he  is  ready  to  infer,  as  is  proved  in  geometry, 
that  the  volume  of  a  pyramid  is  equal  to  one-third  of  the 
volume  of  a  right  prism  of  equal  base  and  altitude.  The 
volume  of  a  cone  may  be  compared  with  that  of  a  cylinder 
of  equal  base  and  altitude  by  the  same  method. 

Measurement  of  the  Sphere 

Pupils  are  usually  interested  in  the  experiment  to  de- 
termine the  area  of  the  surface  of  a  sphere.  Cut  a  wooden 
ball  through  the  center  by  a  plane.  Place  a  tack  at  the 
center  of  the  sphere  and  let  a  pupil  wind  tape  about  this 
sufficient  to  just  cover  the  surface  of  the  great  circle.  Let 


312  HOW  TO  TEACH  AEITHMETIC 

the  pupil  wind  sufficient  tape  about  the  hemisphere  to  just 
cover  its  curved  surface  and  compare  the  amount  of  tape 
required  to  cover  the  two  areas.  If  the  work  has  been 
carefully  done,  it  will  be  found  that  twice  as  much  tape 
is  required  to  cover  the  hemisphere  as  to  cover  the  great 
circle  of  the  sphere  The  number  of  units  in  the  area  of 
the  great  circle  of  the  sphere  is  know  to  be  ?rr2,  where  "r" 
represents  the  number  of  units  in  the  radius,  —  therefore, 
the  number  of  units  in  the  curved  surface  of  the  hem- 
isphere is  2,-jrr2  and  the  surface  of  the  entire  sphere  is 


Use  of  Literal  Representation 

Most  pupils  see  the  advantage  of  letting  "s"  represent 
the  number  of  units  in  the  side  and  "a"  the  number  of 
units  in  the  altitude.  If  the  pupils  are  required  to  use 
these  and  other  abbreviations  in  the  work  of  the  eighth 
grade  a  large  saving  of  time  may  result. 

Problems 

It  is  now  proposed  to  consider  in  some  detail  a  few  prob- 
lems somewhat  more  difficult  than  most  of  those  in  the 
arithmetic  of  the  grades.  Such  a  consideration  should  be 
valuable  from  the  teacher's  standpoint,  because  it  may 
extend  the  margin  of  scholarship  somewhat  and  because  of 
methods  that  are  suggested  for  the  solution  of  such  prob- 
lems. The  problems  that  follow  are  of  a  type  that  are  a 
good  test  of  the  ability  of  the  pupil  to  image  clearly. 

1.  Compare  the  area  of  a  square  with  that  of  the  largest 
circle  that  may  be  cut  from  it. 

In  solving  such  a  problem  it  is  necessary  that  the  pupil 
should  image  the  figure  clearly  and  should  see  what  dimen- 
sions the  circle  and  the  square  have  in  common.  If  the 


MENSURATION  313 

pupil  is  not  able  to  image  this  relationship  readily  the  figure 
may  be  drawn.  It  is  seen  that  the  side  of  the  square  is 
.  equal  in  length  to  the  diameter  of  the  circle.  Here  then  is 
a  basis  for  comparing  their  areas. 

Let  r  =  the  number  of  units  in  the  radius  of  the  circle. 

Then  2r  =  the  number  of  units  in  the  diameter  of  the 
circle. 

Therefore,  2r  =  the  number  of  units  in  the  side  of  the 
square. 

The  area  of  the  circle  is  -n-r2. 

The  area  of  the  square  is  (2r)2  or  4r2. 

Therefore  the  area  of  the  circle  is  to  the  area  of  the 

o  •         A  •>      7rr2      ^ 
square  as  ?rr2  is  to  4r2  or  —  -  =  -—  • 

4r~      4 

fjm 

Since  *  =  *£  we  have  v  ^  =||  or  \\. 

Therefore  the  area  of  the  circle  is  \\  of  the  area  of  the 
square.  This  result  is  true  irrespective  of  how  large  or  how 
small  the  given  square  may  be. 

2.  The  problem  to  compare  the  area  of  a  circle  with  the 
largest  possible  square  that  can  be  cut  from  it,  is  slightly 
more  difficult.  In  this  problem  the  diameter  of  the  circle 
is  seen  to  be  equal  to  the  diagonal  of  the  square. 

Let  r  -  the  number  of  units  in  the  radius  of  the  circle. 

Let  2r  =  the  number  of  units  in  the  diagonal  of  the  square. 

Since  the  number  of  units  in  the  diagonal  of  the  square 
=  2r,  the  number  of  units  in  the  square  of  one  side  is  easily 
found  to  be  2r2.  This  is  the  area  of  the  square. 

The  area  of  the  circle  is  ?rr2. 

The  area  of  the  circle  is  to  the  area  of  the  square  as 


___ 

2r2~        2       -if"     7- 

The   fact   that   such   problems   are   considered   also   in 
geometry  should  not  deter  the  teacher  from  considering 


314  HOW  TO  TEACH  AEITHMETIC 

them  in  the  more  advanced  work  in  mensuration  in  the 
grades. 

3.  Compare  the  volume  of  a  cube  with  the  volume  of  the 
largest  possible  sphere  that  can  be  cut  from  the  cube. 
It  is  easily  seen  that  the  length  of  the  diameter  of  the 
sphere  is  equal  to  the  length  of  the  edge  of  the  cube. 

Let  r  =  the  number  of  units  in  the  radius  of  the  sphere. 

47IT3 

Therefore  —  -  —  =  the  number  of  units  in  the  volume  of 

o 

the  sphere. 

2r  =  the  number  of  units  in  the  edge  of  the  cube. 

Therefore  8^  =  the  number  of  units  in  the  volume  of 
the  cube. 

The  ratio  of  the  volume  of  the  cube  to  the  volume  of  the 
sphere  is  therefore, 


The  volume  of  the  cube  is,  therefore  ||  of  the  volume  of 
the  sphere.  It  would  be  an  easy  problem  to  compare  the 
surface  of  the  cube  with  that  of  the  sphere. 

It  is  sometimes  well  to  put  the  best  pupils  of  the  class 
on  their  mettle  and  such  problems  are  well  adapted  to  this 
purpose. 

Observational  Geometry 

There  is  a  tendency  to  curtail  the  work  in  mensuration, 
which  formerly  consisted  largely  of  definitions,  formulas 
and  problems,  and  to  include  some  elementary  geometry 
in  the  course  of  the  seventh  and  eighth  grades.  The  type 
of  work  introduced  is  variously  known  as  constructional, 
observational,  inventional,  intuitional,  or  concrete  geom- 
etry. Its  purpose  is  to  acquaint  the  pupil  with  the  more 
important  geometrical  concepts,  to  train  the  eye  and 


MENSURATION  315 

the  hand  in  the  use  of  the  straight  edge,  compasses,  tri- 
angle and  protractor,  and  to  develop  the  powers  of  obser- 
vation and  intuition  as  applied  to  geometrical  forms. 

It  is  desirable  that  pupils  who  do  not  enter  high  school 
should  have  some  knowledge  of  the  fundamental  concepts  of 
geometry.  An  introductory  course  in  geometry  prepares 
those  pupils  who  enter  high  school  for  the  more  formal 
study  of  the  subject. 

Some  work  in  introductory  geometry  is  now  given  in 
about  25%  of  the  larger  schools  of  the  country.  The  in- 
struction is  sometimes  made  an  integral  part  of  the  course 
in  arithmetic  and  is  included  under  the  topic  of  mensura- 
tion ;  in  some  schools  one  or  more  periods  a  week  are  regu- 
larly devoted  to  the  work  in  the  seventh  and  eighth  grades 
and  the  connection  with  mensuration  is  not  so  immediate. 

In  many  of  the  European  schools  the  teacher  of  mathe- 
matics is  also  the  teacher  of  drawing,  and  work  of  the 
kind  referred  to  above  is  made  a  part  of  the  instruction  in 
drawing.  It  has  been  suggested  that  one  reason  for  the 
slowness  of  the  introduction  of  this  type  of  work  into  Ameri- 
can schools  is  that  some  work  in  ' l  mechanical ' '  drawing 
is  usually  given  in  the  art  courses.  When  such  has  been 
the  case  emphasis  has  usually  been  placed  upon  artistic 
results  rather  than  upon  the  training  of  the  power  of  ob- 
servation and  generalization,  because  our  art  teachers  do 
not  usually  emphasize  the  subject  from  the  point  of  view 
of  mathematics.  The  schools  in  which  these  constructional 
exercises  are  taught  usually  attempt  to  correlate  the  work 
as  closely  as  seems  practicable  with  the  course  in  manual 
training,1 

In  several  of  the  German  states  the  course  in  arithmetic 
includes  the  study  of  some  of  the  simple  geometric  forms 

iSee  "Mathematics  in  Elem.  Sch.  of  U.  S."  Eeport  to  Inter- 
national Commission,  p.  136. 


316  HOW  TO  TEACH  ARITHMETIC 

and  constructions  as  early  as  the  fifth  and  sixth  years. 
The  pupils  become  familiar  with  the  fundamental  prop- 
erties of  straight  lines,  angles,  triangles,  quadrilaterals, 
polygons,  and  circles,  and  models  of  some  of  the  simpler 
geometrical  solids  are  constructed. 

Simple  and  easily  available  material  for  the  study  of 
elementary  geometrical  forms  is  abundant,  and  the  course 
in  mathematics  in  the  grades  may  be  both  enriched  and 
vitalized  by  a  judicious  use  of  this  material.  Many  of 
the  most  important  theorems  of  geometry  may  be  in- 
tuitively established  by  the  use  of  constructions  and  meas- 
urement. A  better  insight  is  also  gotten  into  most  of  the 
rules  and  formulas  of  mensuration. 

Pupils  of  the  grammar  grades  should  be  taught  the 
meaning  of  such  terms  as  perpendicular,  right  angle,  par- 
allel, etc.  They  should  be  required  to  construct  at  the 
blackboard  and  by  the  use  of  paper  or  cardboard  the  right, 
isosceles,  equilateral  and  scalene  triangles  and  by  means 
of  cutting  out,  tracing,  applying  and  the  use  of  the  pro- 
tractor they  should  discover  experimentally  the  elementary 
truths  about  congruency  of  triangles  and  parallelograms, 
and  the  size  and  relation  of  angles  in  the  various  tri- 
angles. The  formulas  for  the  area  of  the  triangle,  par- 
allelogram and  trapezoid  should  be  discovered  through 
drawing,  paper  cutting  or  by  means  of  tracing.  In  some 
schools  plotting  paper  is  extensively  used  as  a  means  of 
estimating  areas. 

Congruent  and  Similar  Figures 

The  properties  of  congruent  and  of  similar  figures  may  be 
used  by  the  pupils  in  a  variety  of  interesting  and  instruct- 
ive ways.  Ask  the  pupils  to  devise  two  or  three  ways  for 
determining  the  height  of  a  tree  or  a  building  on  the 
school  grounds.  The  following  methods  are  suggested:  (a) 


MENSURATION 


317 


By  comparing  the  length  of  the  shadow  of  a  tree  or  build- 
ing with  that  of  a  pole  of  known  height,  (b)  By  use  of  an 
isosceles  right  triangle.  If  AB=BE,  then  AC  =  CD,  there- 
fore to  find  CD  measure  AC.  (c)  By  drawing  to  scale. 


Let  pupils  determine  the  distance  between  two  inacces* 
sible  points  by  use  of  congruent  triangles. 

Set  up  a  pole  at  0  and 
measure  the  distances 
OX  and  OY.  On  'a 
line  with  X  and  0  put 
a  stake  at  R  such  that 
the  distance  OX  =  OR. 
On  a  line  with  Y  and* 
0  put  a  stake  at  M 
such  that  OY  =  OM. 
Measure  the  distance 
RM.  This  will  be  equal 
to  XY,  which  is  the  re- 
quired distance. 

Ask  pupils  to  devise 
methods    of   determin- 
ing the  width  of  a  river  without  crossing  it. 


318 


HOW  TO  TEACH  AEITHMETIC 


The  following  methods  may  be  used: 
First  Method.    To  determine  the  distance  A  B. 
A  pupil  who  has  an  isosceles  right  triangle  may  walk 
along  the  bank  sighting  along  the  hypotenuse  A  C  and 


B 


C 


holding  one  arm  parallel  to  the  line  B  C,  until  he  reaches 
a  point,  C  such  that  the  point  A  on  the  opposite  bank  from 
B  is  seen.  The  distance  B  C  is  then  equal  to  A  B. 

Second  Method.    To  determine  the  length  of  A  B. 

Mark  off  a  line  B  M  perpendicular  to  A  B  and  place 
states  at  points  B  and  M.  Erect  perpendicular  M  R  of  a 


convenient  length  to  line  M  B.    Place  the  stake  at  C  where 
the  line  B  M  intersects  the  imaginary  line  A  E.     The  dis- 

A    T>  T>    C\ 

tance  A  B  is  then  found  from  the  proportion 


^  -  TVFTT  * 


in  which  M  E7  B  C  and  M  C  are  known. 


CHAPTEE  XXII 
GRAPHS 

Newspapers,  popular  magazines  and  trade  journals  make 
frequent  use  of  graphs  in  order  to  make  clear  the  rela- 
tions between  magnitudes.  It  is  desirable  that  some  in- 
struction in  this  subject  should  be  given  in  the  schools. 
The  pupil  of  the  seventh  or  eighth  grade  is  more  or  less 
familiar  with  the  practice  of  representing  the  relative  sizes 
of  armies  and  navies  by  men  of  various  heights.  He  has 
seen  lines  of  various  lengths  used  to  represent  the  growth 
of  the  population  and  he  knows  that  curves  are  sometimes 
used  to  represent  variations  in  temperature  or  fluctua- 
tions in  the  prices  of  certain  commodities.  In  all  of  the 
more  progressive  European  countries  the  subject  of  graphs 
is  systematically  taught  before  the  pupil  enters  what  cor- 
responds to  our  secondary  school.  Teachers  in  this  coun- 
try are  beginning  to  appreciate  the  fact  that  the  graph 
may  be  used  to  advantage  to  illuminate  certain  topics  and 
to  emphasize  the  relations  between  various  magnitudes. 
The  graph  is  an  effective  method  of  impressing  upon  the 
pupil  relations  that  might  otherwise  be  obscure.  The 
underlying  principles  are  easily  understood,  and  some  con- 
sideration should  be  given  to  the  subject  in  the  sixth, 
seventh,  or  eighth  grades.  Rates  of  increase  and  decrease 
are  always  shown  more  clearly  by  the  use  of  graphs  than 
by  tables  of  statistics.  When  two  or  more  graphs  are 
drawn  on  the  same  scale  and  with  the  same  axes,  compari- 
son of  the  variations  can  be  readily  made.  The  graph 
should  not  be  taught  as  an  end  in  itself,  but  as  a  means 

319 


320 


HOW  TO  TEACH  AEITHMETIC 


to  an  end.    It  should  be  used  to  supplement,  not  to  supplant 
analyses. 

The  graph  is  a  natural  place  for  the  introduction  of  the 
function  idea  which  is  extensively  used  in  higher  mathe- 
matics. Much  more  emphasis  is  placed  upon  the  idea  of 
function  in  the  elementary  and  secondary  schools  of  Europe 
than  in  this  country.  It  is  probable  that  within  a  few 
years  we  shall  place  more  emphasis  upon  this  idea  in  our 
schools.  When  magnitudes  are  so  related  that  any  varia- 
tion of  one  causes  a  variation  of  the  other,  each  is  said  to 
be  a  function  of  the  other.  For  example,  the  distance 
that  a  train  will  run  in  a  certain  time  is  a  function  of  the 
rate,  because  the  distance  varies  as  the  rate  varies.  The 
interest  at  a  given  rate  is  a  function  of  the  principal  and  of 
the  time,  because  as  these  vary  the  interest  varies.  The  area 
of  a  triangle  is  a  function  of  its  base  and  its  altitude,  and 
the  area  of  a  circle  is  a  function  of  its  radius. 


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GRAPHS 


321 


The  topic  may  be  introduced  by  showing  the  pupil  how 
he  may  represent  his  daily  record  in  a  given  subject. 

The  preceding  graph  represents  the  achievement  of  a 
pupil  in  solving  some  problems  in  arithmetic  in  a  limited 
time.  The  graph  shows  the  number  solved  correctly  for 
ten  consecutive  days.  On  the  horizontal  axis  "OX"  the 
days  are  represented,  and  on  the  vertical  axis  "OY"  the 
number  of  problems  solved  correctly  is  shown.  The  graph 
shows  that  on  the  second  day  six  problems  were  solved; 
on  the  third  day,  five ;  on  the  ninth  day,  eight : 

Most  pupils  like  to  graph  the  daily  variations  in  tem- 
perature. Interesting  graphs  may  be  made  showing  the 
fluctuations  in  temperature  at  a  given  hour — for  example, 
9  A.M. — for  several  consecutive  days.  The  graph  below 
illustrates  this. 


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1       2      3      4      5       6       7      8      9      10     11    12     13     U 
Days 

At  9  A.  M.  on  the  first  day  the  temperature  was  58° ;  on 
the  sixth  day  it  was  54° ;  on  the  llth  day  it  was  66°. 
Ask  the  pupils  to  graph  the  growth  of  the  population  of 


322  HOW  TO  TEACH  ARITHMETIC 

the  city  or  county  in  which  they  live,  and  then,  using  the 
same  axes  and  the  same  scale,  graph  the  growth  in  school 
population  for  the  city  or  county.  Pupils  will  be  inter- 
ested in  comparing  these  graphs.  Similar  graphs  may  be 
made  for  the  state  or  the  nation.  Statistics  for  such  graphs 
may  be  readily  secured  in  most  communities. 

Numerous  interesting  facts  whose  relations  are  empha- 
sized by  the  use  of  graphs  may  be  easily  found  by  teacher 
and  pupil.  Most  pupils  from  rural  communities  will  be 
interested  to  graph  the  yearly  yield  of  wheat,  corn,  oats, 
barley,  potatoes. 

County,  state,  and  government  reports  furnish  numerous 
statistics  that  are  of  interest  to  pupils. 

If  the  school  engages  in  athletic  contests,  some  of  the 
pupils  who  are  interested  in  these  activities  may  be  asked 
to  graph  the  results  of  a  series  of  contests.  The  number  of 
points  or  of  goals  scored  by  each  player  or  each  Jeam  may 
be  graphed. 

Some  pupils  in  the  school  may  be  interested  to  keep  a 
graphic  record  of  the  attendance  and  to  compare  this 
graph  with  the  graph  of  various  facts  that  affect  the 
attendance,  such  as  temperature,  rainy  days,  number  of 
cases  of  sickness  in  the  community,  etc. 

Ask  the  pupils  to  make  a  graph  for  the  simple  interest 
on  $1  at  various  rates,  and  require  them  to  use  this  graph 
in  solving  several  problems.  Compare  the  graph  for  simple 
and  for  compound  interest  on  a  given  principal,  at  a  given 
rate,  for  6,  7,  or  8  years. 

Show  the  pupils  how  the  graph  may  be  used  in  the  solu- 
tion of  problems.  For  example :  A  and  B  start  from  the 
same  place  and  travel  in  the  same  direction.  A  travels  at 
the  rate  of  4  miles  an  hour  and  B  at  the  rate  of  6  miles 
an  hour.  If  A  has  three  hours  start,  in  how  many  hours 
will  B  overtake  A? 


GRAPHS 


323 


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The  graph  shows  that  at  the  end  of  the  first  hour  A  is  at 
C  and  his  path  is  OC.  At  the  end  of  the  second  hour  he 
is  at  D.  One  hour  after  B  starts  he  is  at  E.  B  overtakes 
A  when  his  path  crosses  that  of  A.  Determine  by  use  of  a 
graph  how  many  hours  this  is  after  B  starts,  and  how  far 
each  has  traveled. 

Eequest  the  pupils  to  graph  some  facts  in  which  they 
are  interested,  and  from  these  graphs  ask  other  pupils  to 
state  some  of  the  facts. 

NOTE. — Teachers  who  wish  an  elementary  book  on  the  subject  of 
graphs  are  referred  to  Auerbach  's  ' '  An  Elementary  Course  in  Graphic 
Mathematics. ' '  Published  by  Allyn  &  Bacon. 


CHAPTER  XXIII 

SHORT  CUTS 

The  "why"  is  of  importance  in  the  teaching  of  arithme- 
tic, especially  after  the  fifth  or  sixth  school  years.  In  the 
later  years  of  the  grammar  grades  the  "how"  should  re- 
ceive no  less  emphasis  than  in  the  earlier  year,  but  the 
"why"  should  receive  gradually  increasing  emphasis.  To 
many  pupils  arithmetic  is  composed  of  numerous,  dog- 
matically stated  rules  and  scores  of  definitions;  they  have 
derived  but  little  from  that  richer  part  of  arithmetic 
which  emphasizes  the  thought  side. 

When  Short  Cuts  Should  be  Introduced 

However,  in  the  upper  grammar  grades  there  are  times 
when  "the  how" — the  mere  doing  of  a  thing  in  the  cor- 
rect way — should  receive  no  little  emphasis.  After  the 
fundamental  operations  have  been  mastered  and  the  gen- 
eral underlying  principles  of  arithmetic  are  understood, 
certain  short  cuts  may  be  introduced  with  profit. 

It  is  not  intended  that  the  teaching  of  short  cuts  should 
be  postponed  until  the  arithmetic  of  the  grades  has  been 
practically  completed,  but  it  is  best  that  before  a  pupil 
is  taught  a  short  cut  he  should  know  the  longer  method  of 
securing  the  same  result.  The  longer  method  is  usually 
more  easily  explained  and  is  more  likely  to  appeal  to 
the  understanding  of  the  pupil.  If  arithmetic  were  taught 
merely  as  a  tool  to  be  used  in  order  to  obtain  correct  re- 
sults in  computation,  the  short  cut  should  be  introduced 
earlier  than  is  here  advocated. 

324 


SHORT  CUTS  325 

How  to  Teach  Short  Cuts 

Short  cuts,  if  properly  presented,  will  engender  much 
interest  and  enthusiasm  in  the  subject  and  will  save  time 
in  future  computations.  In  teaching  short  cuts  the  teacher 
should  carefully  emphasize  the  fact  that  there  is  not  a 
single  short  process  in  grammar  school  arithmetic  that 
cannot  be  explained.  Some  of  them  are  easily  proved  by 
the  use  of  elementary  algebra.  It  is  neither  necessary  nor 
desirable  that  many  of  these  proofs  should  be  presented 
to  a  class,  but  the  confidence  of  the  pupil  will  be  increased, 
if  he  is  assured  that  a  given  short  cut  can  be  readily 
explained  and  that  it  is  not  due  to  the  so-called  "mystery 
of  numbers. " 

A  short  cut  becomes  really  valuable  to  a  pupil  after  he 
has  such  a  mastery  of  it  that  he  recognizes  it  under  what- 
ever conditions  it  may  be  presented.  One  that  has  not 
been  thoroughly  mastered  and  impressed  upon  the  mind 
by  frequent  drill  is  of  comparatively  little  value.  When 
an  occasion  arises  in  which  the  short  cut  may  be  used  to 
advantage,  the  mind  trained  to  use  the  longer  method 
fails  to  utilize  the  short  cut.  A  few  of  these  short 
processes  thoroughly  mastered  are  of  more  value  than 
many  of  them  not  well  enough  known  to  be  used  upon  the 
proper  occasion.  Any  short  cut  that  is  learned  me- 
chanically will  be  forgotten  within  a  short  time  after  the 
drill  upon  it  ceases,  unless  the  pupil  has  such  a  thorough 
mastery  of  it  that  he  uses  it  almost  as  mechanically  as  he 
should  use  the  facts  of  the  multiplication  table,  and  unless 
his  work  is  of  such  a  nature  that  frequent  opportunity  to 
use  the  short  cut  is  presented.  Choose  a  few  of  the  short 
cuts  which  seem  especially  valuable,  master  them  thor- 
oughly and  be  continually  upon  the  lookout  for  an  oppor- 
tunity to  use  them.  When  a  few  of  them  have  been  so 


326  HOW  TO  TEACH  AKITHMETIG 

mastered  that  their  application  has  become  more  or  less 
mechanical  a  few  more  should  be  taken  up  in  the  same 
way.  The  importance  of  short  cuts  can  be  easily  over- 
emphasized. It  would  be  unwise  for  any  teacher  to  at- 
tempt to  have  the  class  master  all  the  short  processes  that 
are  suggested  here. 

Short  Cuts  in  Multiplication 

Multiplication  is  a  short  cut  for  addition.  When  it  is 
required  to  multiply  47  by  18  the  question  really  is,  "what 
is  the  result  when  47  is  used  18  times  as  an  addend  ? ' '  Our 
mastery  of  the  multiplication  table  and  our  knowledge  of 
place  value  enables  us  to  obtain  the  required  result  by  a 
shorter  process  than  addition.  The  example  just  cited 
could  not  be  so  quickly  worked  by  one  who  is  not  familiar 
with  the  multiplication  table  unless  he  used  some  mechan- 
ical device  to  aid  him  in  the  computation  The  Eussian 
peasant  who  can  add  and  can  multiply  and  divide  by  2 
could  work  the  above  example,  but  he  could  not  work  it 
in  the  way  most  familiar  to  us.  He  would  proceed  as 
follows : 

._      ..^  Each    number   in    the    first    column    is 

~~       ~fi  divided  by  2 ;  if  the  quotient  is  not  an  in- 

.. ..       _~  teger  only  the  integral  part  of  the  quotient 

.   "      ^  ..  is  taken;  this  division  is  continued  until 

the  quotient  1  is  reached.     In  the  second 

column  each  number  is  multiplied  by  2; 

this  is  continued  until  as  many  numbers 

have  been  obtained  as  in  the  first  column.    All  numbers  in 

the  second  column,  except  those  which  stand  opposite  even 

numbers  in  the  first  column  are  added.     The  sum  thus 

obtained  is  the  required  product;  i.e.,  576  + 144  +  72 -f  36  + 

18=846,  or  47x18.     It  is  apparent  that  the  term  "short 

cut"  is  a  relative  term.     If  our  method  of  multiplying 


SHORT  CUTS  327 

could  be  taught  to  a  Russian  peasant  it  would  be  a  val- 
uable short  cut  for  him. 

Professional  mathematicians  use  numerous  short  cuts 
for  obtaining  results.  Some  of  these,  such  as  tables  of 
logarithms;  tables  of  squares  and  cubes,  and  of  square 
roots  and  cube  roots,  save  an  enormous  amount  of  time  in 
computation,  but  not  all  of  these  can  be  used  to  advantage 
by  a  pupil  in  the  upper  grammar  grades. 

Attention  will  be  directed  first  to  some  short  cuts  that 
can  be  used  to  advantage  in  multiplication. 

To  square  any  number  ending  in  5.  Multiply  the  num- 
ber of  tens  by  one  more  than  itself  and  annex  25.  For 
example,  to  square  35.  The  number  of  tens  is  3.  Multiply 
this  by  1  more  than  itself  and  the  product  is  12.  Annex 
25  and  the  final  result  is  1225.  By  the  same  rule  the  square 
of  45  is  found  to  2025;  and  the  square  of  75  is  5625. 
1052- 11025. 

The  above  is  obviously  applicable,  with  modifica- 
tions, to  examples  like  the  following:  (7^)2,  (9^)2,  (4^)2. 
But  it  is  easier  to  work  such  examples  by  the  following 
rule:  Multiply  the  integer  by  one  more  than  itself  and 
add  1.  Thus,  (7±)2=  (8x7)  +  i  =  5.6J.  (9±)2=  (10x9) "+ 
i  =  90£.  (4J)2=  (5x4)  +i  =  20i  "  It  is  evident  that  the  i 
may  be  written  as  .25. 

Required  to  multiply  two  numbers  whose  ten's  digits 
are  the  same  and  whose  unit's  digits  add  to  make  10.  For 
example,  47x43,  or  84x86.  Multiply  the  ten's  digit  by 
one  more  than  itself  and  annex  the  product  of  the  unit's 
digits.  Thus,  47x43  equal  2021.  84x86  equal  7224. 
13x17  equal  221.  Whenever  the  product  of  the  unit's 
digits  is  a  one  digit  number,  a  zero  must  be  put  in  the 
tens'  place.  For  example,  if  81  is  to  be  multiplied  by  89, 
the  product  of  the  unit's  digits  (1  and  9)  is  a  one  digit 
number,  hence  the  final  result  is  7209.  Similarly  the  prod- 


328  HOW  TO  TEACH  AE1THMETIC 

uct  of  51  and  59  equal  3009.  The  rule  for  squaring  a 
number  ending  in  5  is  evidently  a  special  case  of  the  rule 
just  cited. 

To  multiply  two  numbers  between  10  and  20.  For  ex- 
ample, 17x15,  or  14x17,  or  19x16. 

To  either  of  the  numbers  add  the  unit's  digit  of  the 
other.  Annex  a  cipher  to  this  result  and  add  the  product 
of  the  unit's  digits.  Example,  17x15.  17  +  5  equal  22  (or 
15  +  7  equal  22).  Annex  a  cipher  and  the  result  is  220. 
To  this  add  the  product  of  the  unit's  digits  and  the  result 
is  220 +  35,  or  255. 

Example,  14x17.  14  +  7  =  21.  Annex  a  cipher  and  the 
result  is  210.  Add  4x 7  to  this,  and  the  result  is  238. 

This  computation  may  always  be  made  orally  with  ease 
and  is  a  valuable  short  cut. 

When  one  factor  is  as  much  greater  than  some  multiple 
of  10  as  the  other  is  less  than  that  same  multiple. 

Example,  28x32.  (32  is  2  more  than  30  and  28  is  2 
less  than  30). 

Example,  46x54.  (54  is  4  more  than  50,  and  46  is  4 
less  than  50). 

Square  the  multiple  of  ten  and  from  this  result  subtract 
the  square  of  the  difference  between  one  of  the  given  num- 
bers and  the  multiple  of  10. 

Example,  28x32.  30  is'the  multiple  of  10  between  these 
numbers.  302  =  900.  The  difference  between  one  of  the 
given  numbers  and  30  is  2.  22  =  4.  900-4  =  896,  which  is 
the  required  product. 

Example,  46x54.     502  =  2500.     42  =  16.     2500-16  =  2484. 

Example,  39x41.     402  =  1600.     12  =  1.     1600-1  =  1599. 

Those  who  understand  the  elements  of  algebra  will  recog- 
nize the  above  short  cut  as  (a-b)  (a  +  b)  =a2-b2,  since 
we  may  express  28x32  as  (30-2)  (30  +  2)  =  302-22=900- 
4  equal  896. 


SHORT  CUTS  329 

v 

The  product  of  any  2  numbers  equals  the  square  of  the 
number  midway  between  them,  minus  the  square  of  half 
their  difference. 

This  comes  under  the  same  algebraic  law  as  the  short 
cut  preceding. 

Example,  17xl3  =  152-4  =  221. 

Find  the  cost  of  19  books  at  13c  each.    Eesult  in  cents  - 


By  the  complement  of  a  number  is  meant  that  number 
which  added  to  the  given  number  makes  the  next  higher 
power  of  10.  Thus  the  complement  of  98  is  2  ;  of  87  it  is 
13,  of  992  it  is  8. 

To  multiply  together  two  numbers  whose  complements 
are  not  large  and  whose  complements  are  computed  from 
the  same  power  of  ten,  proceed  as  follows: 

Example.     98x95.  98     2  =  complement  of  98. 

95  _  5  =  complement  of  95. 
98-5  and  then  annex  10,  (5x2)  93  10  =  required  product. 

From  either  number  (98  or  95)  subtract  the  comple- 
ment of  the  other  number.  To  this  result  annex  the  prod- 
uct of  the  complements. 

Example.     89x93.  89  11  =  complement  of  89. 

93  _  7j=  complement  of  93. 
89-7  and  then  annex  77,  (7  x  11  )  8277  -  required  result. 

Example.     989  x  988.  989  11  =  complement  of  989. 

988  12  =  complement  of  988. 

Subtract  12  from  989  and      977132  =  required  result. 
annex  132,  (11x12). 

When  the  complement  is  computed  from  100  and  the 
product  of  the  complements  is  not  a  two  digit  number  a 
zero  must  be  put  in  tens'  place.  When  the  complement  is 
computed  from  1000  and  the  product  of  the  complements 
is  not  a  three  digit  number,  a  zero  must  be  put  in  hun- 
dreds' place,  or  in  hundreds'  and  and  in  tens'  places. 


330  HOW  TO  TEACH  ARITHMETIC 

Example,  97x99.  The  product  of  the  complements  (3 
and  1)  is  not  a  number  of  two  digits.  The  result  is  9603. 

Example,  998x994-992012  (2x6  is  only  a  two  digit 
number). 

Example,  999x997  =  996003  (1x3  is  only  a  one  digit 
number). 

By  the  supplement  of  a  number  is  meant  that  which  sub- 
tracted from  the  number  makes  the  next  low.er  power  of 
ten.  Thus  the  supplement  of  104  is  4;  the  supplement  of 

111  is  11 ;  the  supplement  of  1007  is  7. 

To  multiply  together  two  numbers  whose  supplements 
are  not  large  and  whose  supplements  are  computed  from 
the  same  power  of  ten,  proceed  as  indicated  below: 

Example,  112x103.  112  12  =  supplement  of  112. 

103  3  =  supplement  of  103. 

112  +  3,  then  annex  36,  (3x12)   11536  =  required  product. 
To  either  number,  112  or  103,  add  the  supplement  of  the 

other.  To  this  result  annex  the  product  of  the  supple- 
ments. 

Example.  117x105.  117  17  =  supplement  of  117. 

105  5  =  supplement  of  105. 
117  +  5,  then  annex  85,  (5x17)  12285  =  required  product. 

When  the  product  of  the  supplements  does  not  give  a 
two  digit  number  a  zero  must  be  put  in  tens'  place  if  the 
supplement  is  computed  from  100.  If  the  supplement  is 
computed  from  1000  and  the  product  of  the  supplements 
is  not  a  three  digit  number,  a  zero  must  be  put  in  the 
hundreds,  or  the  hundreds'  and  the  tens'  places. 

Example,  102x104.  The  product  of  the  supplements 
(2  and  4)  is  a  one  digit  number.  The  result  is  10608. 

Example,  1014x1003  =  1017042. 

After  a  short  drill  the  short  cuts  involving  complements 
and  supplements  can  be  quickly  applied  without  the  com- 
plements or  supplements  being  written  down. 


SHORT  CUTS  331 

The  Elevens  Rule 

Example,  11x4532. 

Write  2  for  the  right  hand  figure  of  the  product.  Add 
the  2  and  3  for  the  next  figure  of  the  product.  Add  3  and 
5  for  the  third  figure  of  the  product ;  add  the  5  and  4  for 
the  fourth  figure  and  write  4  for  the  fifth  figure  of  the 
product.  The  product  is,  therefore,  49852.  The  reason  for 
this  procedure  is  easily  seen  by  considering  the  longer  form 
and  seeing  the  additions  that  must  be  made. 

4532 
11  Similarly  11  x  482  =  5302. 

4532 
4532 
49852 

Aliquot  Parts.  By  the  aliquot  parts  of  a  number  are 
meant  those  n ambers  which  are  contained  in  the  given  num- 
ber without  a  remainder.  Thus  4  is  an  aliquot  part  of  12 ; 
2^  is  an  aliquot  part  of  5.  When  aliquot  parts  are  used  in 
arithmetic  the  given  number  is  usually  ten,  a  hundred,  or 
a  thousand. 

Aliquot  parts  of  a  number  may  frequently  be  used  to 
advantage  in  multiplication. 

Example.  To  multiply  84  by  25  we  may  annex  two 
ciphers  to  the  84  (that  is,  multiply  it  by  100)  and  then 
divide  the  result  by  4,  since  25  =  J  of  100. 

Similarly  to  multiply  a  number  by  16f  we  may  first 
multiply  the  number  by  100  and  divide  the  result  by  6. 

To  multiply  a  number  by  2J  we  may  multiply  the  given 
number  by  10  and  divide  this  result  by  4,  since  2J  equal 
i  of  10. 

The  pupil  should  learn  the  commonly  used  aliquot  parts 
of  10,  100  and  1000,  as  they  can  frequently  be  used  to 
advantage,  as  suggested  above. 


332  HOW  T0  TEACH  ARITHMETIC 

Frequently  used 

aliquot  parts  Frequently  used  aliquot  parts 

of  ten  of  one  hundred 


2J  =  i  of  10              50  =  4  of  100  12£   =  4    of  100 

34  =  4  of  10               33i  =J  of  100  111    =t    of  100 

5   =  4  of  10               25  =i  of  100  10     =  3*0  of  100 

6f  =  foflO               20  =  4  of  100  9^  =  ^  of  100 

16f  =J  of  100  8J   =  A  of  100 

14f  =1  of  100  6i   ^V  of  100 

The  equivalents  of  the  following  parts  of  100  should  also 
be  learned,  f  ,  f  ,  f  ,  f  ,  |,  f  ,  f  ,  f  ,  and  J. 

The  pupil  should  learn  also  the  more  frequently  used 
aliquot  parts  of  1000,  such  as  4,  i,  4,  and  f  . 

If  required  to  find  the  cost  of  42  articles  at  16§  cents 
.each,  the  pupil  should  utilize  his  knowledge  of  aliquot 
parts.  At  $1  each  the  articles  would  cost  $42,  therefore, 
at  $4  each,  the  cost  will  be  4  of  $42,  or  $7. 

If  the  problem  requires  the  cost  of  24  articles  at  37^ 
cents  each  the  pupil  should  be  trained  to  see  that  the  cost 
will  be  just  f  of  $24,  or  $9. 

To  multiply  two  mixed  numbers  when  the  fractions  are 
each  ^  : 

To  the  product  of  the  whole  numbers  add  one-half  the 
sum  of  the  whole  numbers  and  to  this  add  one-fourth. 

Example,  4^x7^.  4x7  =  28.  ^  the  sum  of  the  whole 
numbers  (4+7)  is  5^.  Add  this  to  28  and  to  this  sum  add 
i.  ,  The  final  result  is  33f. 

Similarly,  5^x9  J.    5x9  =  45.    £  of  (5  +  9)  is  7.    45  +  7  + 


CHAPTER  XXIV 
LONGITUDE  AND  TIME 

The  subject  of  Longitude  and  Time  is  common  to  both 
Arithmetic  and  Mathematical  Geography.  In  recent  years 
the  tendency  has  been  to  omit  the  subject  from  courses  in 
arithmetic,  but  it  is  still  taught  in  so  many  schools  that  its 
treatment  is  necessary  here.  Longitude  and  time  is  a  prac- 
tical subject  to  the  navigator  and  to  the  astronomer.  The 
rapid  adoption  of  standard  time  by  most  of  the  civilized 
nations  of  the  world  has  rendered  impractical  the  older 
text-book  problems  in  longitude  and  time.  The  old-style, 
complicated  problems  of  a  generation  ago  are  giving  way 
to  practical  problems  involving  standard  time. 

The  principles  underlying  the  subject  can  be  easily 
explained,  and  no  teacher  should  permit  the  substitution 
of  arbitrary  rules  for  the  application  of  a  few  simple  prin- 
ciples. Many  pupils  of  the  grades  obtain  but  little  of  value 
from  a  study  of  the  subject  because  it  is  presented  in  the 
form  of  rules, 

In  this  discussion  it  is  assumed  that  the  pupil  knows  the 
meaning  of  the  terms  degree,  longitude,  and  meridian.  It 
is  assumed  also  that  he  is  familiar  with  the  units  of  circular 
measure. 

There  is  only  one  equator  from  which  latitude  may  be 
reckoned,  but  longitude  may  be  reckoned  from  any 
meridian.  Prior  to  1844  it  was  the  custom  in  many  coun- 
tries to  use  the  longitude  of  their  capital  as  the  zero  of 
longitude.  Since  1844  the  meridian  running  through  the 
observatory  at  Greenwich,  England,  has  been  most  gener- 

.333 


334  HOW  TO  TEACH  ARITHMETIC 

ally.  used.      Greenwich   is   about   five   miles   southeast   of 
,  London. 

Pupils  will  be  interested  to  know  the  accuracy  with 
which  longitude  can  be  determined  by  a  skillful  astron- 
omer with  refined  instruments.  The  difference  in  longitude 
between  two  cities  on  the  same  continent  can  be  determined 
within  six  or  eight 'yards;  the  difference  in  longitude  be- 
tween Washington  and  Greenwich  is  known  within  about 
three  hundred  feet. 

The  Tables 

The  tables  for  solving  the  problems  of  longitude  and 
time  may  be  developed  either  by  considering  the  real  rota- 
tion of  the  earth  upon  its  axis  or  the  apparent  revolution 
of  the  sun  around  the  earth. 

Every  point  on  the  earth's  surface,  except  the  poles,  is 
carried,  by  the  earth's  rotation,  through  360°  in  24  hours. 

From  this  fact  the  tables  are  developed : 
Since  in  24  hr.  the  earth  turns  through  360°, 
therefore  in  1  hr.  it  turns  through  -^  of  360°  or  15°, 
therefore  in  V  it  turns  through  -favi  15°,  or  -J°,  or  15', 
therefore  in  1"  it  turns  through^ of  15'  or  y,  or  15. " 
Since  the  earth  turns  through  360°  in  24  hr., 

We  may  say: 
therefore  it  turns  through  1°  in  -%^-Q  of  24  hr.,  or  -^  hr.  or 

4  min., 
therefore  it  turns  through  1'  in  -^  of  4  min.,  or  -^  min.  or 

4  sec., 
therefore  it  turns  through  V  in  -fa  of  4  sec.,  or  T^  sec. 

The  mastery  of  the  preceding  tables  is  of  importance 
in  a  detailed  study  of  longitude  and  time.  The  teacher 
should  not  permit  the  pupil  to  use  the  statement  found  in 
many  text-books,  that  15°  =  1  hour.  There  is  no  equality 
between  longitude  and  time  in  the  sense  that  " equality" 


LONGITUDE  AND  TIME  335 

is  used  in  mathematics.  Two  times  4  equals  8  is  one  of 
the  fundamental  facts  of  mathematics,  whereas  15° 
corresponds  to  1  hour  of  time  only  because  there  are  360° 
in  a  circumference  and  because  the  earth  rotates  upon  its 
axis  once  in  24  hours.  If  the  rotation  period  of  the  earth 
were  20  hours  instead  of  24  hours,  15°  would  no  longer 
correspond  to  1  hour;  18°  would  correspond  to  1  hour. 
After  the  facts  of  the  tables  have  been  mastered,  numerous 
easy  problems  based  upon  them  should  be  given.  The  prob- 
lems below  will  suggest  others  to  the  teacher : 

Two  cities  differ  in  longitude  by  30°,  by  45°,  by  60°, 
by  150°,  by  22°.30',  by  7° ;  what  is  the  difference  in  time? 

The  difference  in  time  between  two  cities  is  3  hours,  5 
hours,  1J  hours,  3J  hours;  what  is  the  difference  in 
longitude  ? 

To  Determine  Difference  in  Longitude 

Pupils  are  sometimes  uncertain  whether  the  longitude  of 
two  given  places  should  be  added  or  subtracted  in  order  to 
obtain  the  difference  in  longitude.  This  uncertainty  may 
be  eliminated  by  illustrations  similar  to  those  below.  The 
teacher  should  give  the  illustrations  a  local  setting 
by  substituting  for  A,  B,  and  C  objects  that  are  familiar 
to  the  pupil.  The  distance  from  the  schoolhouse  to  certain 
places  in  opposite  directions  from  it,  or  distances  from 
the  city  in  which  the  school  is  located  to  two  other  places 
in  opposite  directions  from  the  home  city,  may  be  used. 
~B  20  mi.  A  30  mi.  C"  In  this  illustration 
the  cities  A,  B,  and  C  are  in  the  same  straight  line,  A 
being  20  miles  from  B  and  30  miles  from  C.  What  is  the 
distance  from  B  to  C  ?  The  pupil  will  readily  see  that  the 
distance  from  B  to  C  is  20  miles  plus  30  miles,  or  50  miles. 
If  the  distance  from  B  to  C  is  given  as  50  miles,  and  the 
distance  from  A  to  B  as  20  miles,  it  is  evident  that  the 


336  HOW  TO  TEACH  ARITHMETIC 

distance  from  A  to  C  is  50  miles  minus  20  miles,  or  30  miles. 
From  illustrations  similar  to  the  above  the  pupils  can 
formulate  the  rule  that  when  one  place  is  east  and  another 
is  west  of  a  given  third  place,  to  find  the  distance  between 
the  first  two  places  we  add  their  distances  from  the  third 
place.  When  both  are  east  or  both  west  of  the  given  third 
place,  we  subtract  their  distance  from  each  other.  In  the 
illustration  above,  substitute  the  zero  (Greenwich)  merid- 
ian for  city  A,  instead  of  city  B  substitute  20°  west  longi- 
tude, instead  of  city  C  substitute  30°  east  longitude,  and 
we  have  the  principle  fo.r  determining  when  to  add  and 
when  to  subtract  longitudes.  To  find  the  difference  in 
longitude  between  two  given  places,  if  both  places  are 
east  longitude  or  both  are  west,  subtract  the  given  longi- 
tudes ;  if  one  place  is  east  longitude  and  the  other  is  west, 
add  the  given  longitudes.  It  is  now  customary  to  indicate 
west  longitude  as  positive  (+),  and  east  longitude  as  nega- 
tive (-).  Problems  similar  to  the  following  should  be 
given  to  emphasize  this  principle : 

City  A  is  95°  west  and  city  B  is  30°  east.  What  is  their 
difference  in  longitude  ? 

City  A  is  85°  west  and  city  B  is  122°  west.  What  is 
their  difference  in  longitude  ? 

City  M  is  120°  east  and  city  R  is  18°  east.  What  is  their 
difference  in  longitude  ? 

Illustrative  Problems 

If  the  course  of  study  requires  the  solution  of  problems 
in  longitude  and  time,  in  addition  to  those  involving 
standard  time,  they  may  be  solved  as  in  the  illustrations 
which  follow.  Frequent  use  should  be  made  of  a  globe. 
The  longitude  of  the  city  or  the  town  in  which  the  school  is 
located  can  be  approximately  determined  by  careful  meas- 


LONGITUDE  AND  TIME  337 

urements  on  a  good  map,  and  this  longitude  should  be 
made  the  basis  of  a  number  of  problems. 

The  longitude  of  Chicago,  Illinois,  is  87°  36'  42"  W.  The 
longitude  of  "Washington,  D.  C.,  is  77°  2'  W.  When  it  is 
4  P.  M.  local  time  at  Chicago,  what  is  the  local  time  at 
Washington  ? 

The  difference  in  longitude  is  (87°  36'42")  -  (77°  2')  or 
10°34/42//. 

Since  1°  corresponds  to  4  min., 

therefore  10°  corresponds  to  40  min. 

Since  1'  corresponds  to  T*ff  of  a  minute, 

therefore  34'  corresponds  to  2T\  min.,  or  2  min.  16  sec. 

Since  V  corresponds  to-^-of  a  second, 

therefore  42"  corresponds  to  2-Jf  sec.,  or  2f  sec. 

The  total  difference  in  time  is  42  min.  18f  sec. 

Since  Washington  is  east  of  Chicago,  the  time  in  Wasfr 
ington  is  later  than  in  Chicago.  The  time  in  Washington 
is  therefore  42  minutes  18  j-  seconds  after  4  p.  M. 

Pupils  sometimes  have  difficulty  in  determining  which  of 
two  places  has  the  later  time  at  a  given  moment.  This 
difficulty  may  be  eliminated  by  such  questions  as:  Sup- 
pose it  is  noon  where  you  live ;  is  the  sun  also  on  the 
meridian  of  a  city  30°  west  of  where  you  live?  How  long 
before  it  will  be  on  this  meridian  ?  The  earth  rotates  from 
west  to  east;  how  long  will  it  take  a  given  point  on  the 
earth  to  rotate  through  30°  ?  By  questions  similar  to  this, 
the  idea  is  developed  that  of  any  two  places,  the  one 
farther  west,  measured  on  the  shorter  arc,  has  the  slower 
time. 

The  longitude  of  New  York  City  is  73°  58' 26"  W.,  and 
that  of  Paris,  France,  is  2°  20'  15"  E.  When  it  is  3  :40  A.  M. 
local  time  in  New  York  City,  what  is  the  local  time  in  Paris  ? 

The  difference  in  longitude  is  (73°  58'  26")  +  (2°  20'  15") 


338  HOW  TO  TEACH.  ARITHMETIC 

or  76°  18' 41".  By  use  of  the  tables  as  in  the  preceding 
problem,  we  find  that  76°  18' 41"  corresponds  to  a  differ- 
ence in  time  of  5  hrs.  5  min.  14^  sec.  Since  Paris  is  east 
of  New  York,  the  local  time  in  Paris  is  later  than  in  New 
York.  Therefore  the  time 'in  Paris  is  8  hrs.  45  min.  14}^ 
sec.  A.M. 

Some  text-books  still  use  an  incorrect  form  in  solving  a 
problem  like  the  one  above : 

15)76  18'  41" 


5  hr.     ,    5  min.         14  J--J-  sec. 

Such  a  form  is  inaccurate  and  should  not  be  tolerated. 
If  a  teacher  advocates  such  a  form  because  it  is  brief,  he 
should  recall  the  fact  that  brevity  is  not  the  thing  to  be 
chiefly  sought  in  such  work,  and,  further,  that  those  whose 
business  necessitates  the  solving  of  problems  involving 
longitude  afod  time,  make  their  computations  by  the  aid 
of  longitude  tables  and  not  by  the  inaccurate  method 
described  above. 

The  longitude  of  Albany,  New  York,  is  73°  44' 48". 
When  it  is  2  p.  M.  at  Albany  it  is  20  min.  4  sec.  past  1  P.  M. 
at  another  city.  Find  the  longitude  of  the  second  city. 

The  difference  in  time  between  the  two  cities  is  39  min. 
56  sec.  A  difference  of  39  min.  56  sec.  corresponds  to  a 
difference  of  9°  59'  in  longitude. 

The  second  city  must  be  west  of  Albany,  since  it  has 
earlier  local  time.  The  longitude  of  the  second  city  is 
(73°  44'  48")  +  (9°  59')  or  83°  43'  48".  (This  is  the  Ion- 
gitude  of  Ann  Arbor,  Michigan.) 

Standard  Time 

After  the  relation  between  longitude  and  local  time  has 
been  established,  the  subject  of  standard  time  should  be 


LONGITUDE  AND  TIME  339 

considered.  All  places  in  the  same  longitude  have  exactly 
the  same  local  time  at  any  given  moment.  All  places  in 
different  longitudes  have  different  local  time  at  any  given 
moment.  At  the  equator  1  minute  of  time  corresponds  to 
approximately  17  miles  in  longitude,  and  in  latitude  40°  to 
approximately  13  miles.  It  is  evident  that  if  every  station 
along  a  railway  extending  east  and  west  should  keep  its 
own  local  time  there  would  be  great  confusion  and  the 
chance  for  accidents  would  be  greatly  increased.  To  avoid 
such  confusion  the  railways  of  the  United  States  in  1883 
proposed  a  system  of  time  that  has  since  been  adopted 
by  most  of  the  civilized  nations  of  the  world.  The  system 
is  called  "Standard  Time"  because  the  time  at  any  given 
place  is  considered  to  be  the  same  as  the  time  at  a  certain 
standard  meridian.  In  the  United  States  these  are  four 
standard  meridians.  These  meridians  are  the  75th,  90th, 
105th,  and  120th.  The  country  (approximately  7^°  wide) 
on  each  side  of  these  meridians  is  considered  as  a  time  belt, 
and  every  place  in  a  given  belt  uses  the  time  of  its  merid- 
ian. The  belt  which  uses  the  time  of  the  75th  meridian  is 
called  the  eastern  belt ;  the  90th  meridian,  the  central  belt ; 
the  105th,  the  mountain  belt;  and  the  120th  meridian,  the 
Pacific  belt.  It  should  be  noted  that  the  standard  meridians 
are  15°  apart.  Since  15°  correspond  to  1  hour,  it  follows 
that  the  standard  time  in  two  consecutive  time  belts  always 
differs  by  exactly  one  hour,  the  belt  farther  east  always 
having  the  later  time.  The  time  belts  are  not  exactly  7|-° 
wide  on  each  side  of  the  standard  meridian.  The  belts 
were  originally  proposed  by  the  railways  for  their  own 
'convenience,  and  the  division  terminals  of  the  railways  are 
not  always  just  half  way  between  two  standard  meridians. 
The  line  of  division  between  two  adjacent  belts  usually 
passes  through  the  various  railway  terminals  of  the  vicinity. 
Every  place  in  a  given  time  belt  has  exactly  the  same 


340  HOW  TO  TEACH  AEITHMETIC 

standard  time  as  any  other  place  in  the  same  belt,  and  its 
time  is  just  one  hour  earlier  than  the  time  of  a  place  in 
the  belt  next  to  the  east,  and  one  hour  later  than  that  of 
any  place  in  the  belt  next  to  the  west. 

By  consulting  the  proper  official  railway  guides,  a 
teacher  may  find  out  the  exact  points  where  changes  of 
time  are  made.  The  following  are  some  of  the  principal 
division  points  between  the  time  belts :  Buffalo,  New  York ; 
Pittsburgh,  Pennsylvania ;  Wheeling  and  Huntington,  West 
Virginia;  Atlanta,  Georgia;  and  Charleston,  South  Caro- 
lina, lie  on  the  boundary  between  the  Eastern  and  Central 
belts;  while  Mandan,  South  Dakota;  North  Platte, 
Nebraska;  Dodge  City,  Kansas;  and  El  Paso,  Texas,  lie 
on  the  boundary  between  the  Central  and  Mountain  belts. 

The  75th  meridian,  which  is  the  standard  for  the  Eastern 
time  belt,  passes  approximately  through  Philadelphia, 
Pennsylvania.  The  90th  meridian  passes  approximately 
through  St.  Louis;  the  105th  meridian  passes  through 
Denver;  and  the  120th  meridian  passes  about  100  miles 
east  of  San  Francisco. 

Since  the  standard  meridian  of  one  of  the  time  belts 
passes  through  Philadelphia,  Pennsylvania,  it  follows  that 
local  and  standard  time  are  the  same  in  that  city.  Since 
New  York  City  is  east  of  its  standard  meridian,  the  75th, 
it  follows  that  its  local  time  is  faster  than  it  standard  time ; 
while  in  Harrisburg,  Pennsylvania,  which  is  west  of  its 
standard  meridian,  the  75th,  the  local  time  is  slower  than 
the  standard  time.  Except  where  a  time  belt  is  very 
irregular,  the  extreme  difference  between  local  and  stand- 
ard time  is  about  half  an  hour.  If  the  exact  difference 
between  the  local  and  the  standard  time  in  any  locality 
is  known,  the  longitude  can  be  easily  determined.  Suppose 
the  local  time  in  your  city  is  12  min.  30  sec.  faster  than  the 
standard  time.  Since  12  min.  30  sec.  correspond  to  a  differ- 


LONGITUDE  AND  TIME  341 

ence  of  3°  7'  30"  of  longitude,  it  follows  that  the  city  must 
be  3°  7' 30"  east  of  its  standard  meridian.  (It  must  be 
east  of  the  standard  meridian,  since  the  local  time  is  faster 
than  standard  time.)  If  the  city  is  in  the  Central  time 
belt  its  longitude  is  90°  -  (3°  V  30")  or  86°  52'  30".  If  it 
is  in  the  Mountain  time  belt  its  longitude  is  105° -(3° 
7' 30")  or  101°  52' 30".  The  pupils  should  have  access 
to  a  map  showing  the  various  time  belts  in  the  United 
States, 

The  time  belt  whose  principal  meridian  is  60°.  "W.  is 
called  the  Atlantic  belt.  It  embraces  parts  of  Eastern 
Canada.  Great  Britain,  Holland,  and  Belgium  use  as  their 
standard  meridian  0°.  Most  of  the  mid-European  coun- 
tries use  the  meridian  15°  east  as  their  standard.  Bul- 
garia, Roumania,  and  parts  of  Turkey  use  30°  east.  South- 
ern Australia  and  Japan  use  135°  east. 

Problems  in  Standard  Time 

. 

All  problems  involving  standard  time  can  be  solved 
orally.  After  the  general  boundaries  of  the  time  belts 
in  the  United  States  are  known,  questions  like  the  following 
should  be  answered  without  difficulty: 

When  it  is  10  A.M.  Standard  time  in  Omaha,  Nebraska, 
what  is  the  time  in  Denver,  New  York  City,  Detroit,  St. 
Louis,  San  Francisco,  and  Ogden,  Utah? 

When  it  is  2 :35  P.  M.  Standard  time  in  Indianapolis,  what 
is  the  time  in  New  Haven;  Portland,  Oregon;  Columbus, 
Ohio ;  Austin,  Texas ;  Salt  Lake  City ;  and  Los  Angeles  ? 

Since  longitudes  in  the  United  States  are  reckoned  from 
Greenwich,  England,  it  is  evident  that  the  difference  in 
time  between  a  place  in  England  (0°  meridian)  and  a  place 
in  the  United  States  is  easily  determined.  Eastern  time  is 
just  five  hours  slower  than  Greenwich  time.  Central  time 


342  HOW  TO  TEACH  AEITHMETIC 

is  six  hours  slower  than  Greenwich  time,  etc.  When  it  is 
10  A.  M.  at  St.  Louis,  it  is  therefore  six  hours  later,  or  4  P.M., 
in  London.  When  it  is  11  P.M.  Wednesday  in  St.  Paul, 
Minnesota,  it  is  six  hours  later  in  London,  England.  The 
time  in  London  is  therefore  5  A.  M.  Thursday. 

The  above  consideration  indicates  how  it  is  possible  for 
an  event  which  happens  at  2  P.M.  in  London  to  be  given 
in  detail  in  newspapers  in  the  United  States  which  are 
published  at  10  A.  M.  of  the  same  day. 

Most  pupils  will  be  interested  to  determine  the  time  in 
Berlin,  Germany;  St.  Petersburg,  Russia;  and  Tokio, 
Japan,  when  the  school  day  begins  in  the  United  States. 
Ask  the  pupils  what  time  it  is  in  Philadelphia,  London, 
New  Orleans,  San  Francisco,  Berlin,  and  Tokio  at  the 
hour  they  are  reciting. 

The  Date  Line 

The  question  of  the,  international  date  line  belongs  pri- 
marily to  geography,  but  it  may  be  introduced  here.  The 
time  at  that  point  on  the  earth's  surface  which  is  exactly 
180°  east  of  your  school  is  12  hours  later  than  at  your  school, 
since  180°  corresponds  to  12  hours,  and  since  the  time 
must  be  later  as  the  place  is  east.  The  time  at  that  point 
on  the  earth's  surface  which  is  exactly  180°  west  of  your 
school  is  12  hours  earlier  that  at  your  school,  since  180° 
corresponds  to  12  hours,  and  since  the  place  is  west  it  must 
have  earlier  time.  But  the  place  180°  east  of  the  school 
must  be  the  same  as  the  place  180°  west  of  the  school.  It  is 
evident  that  at  this  place  it  cannot  at  the  same  instant  be 
both  twelve  hours  earlier  and  twelve  hours  later  than  the 
time  at  your  school.  This  must  be  true,  however,  unless 
some  fact  has  been  omitted  in  the  calculation.  In  order  to 
rectify  this  apparent  impossibility,  another  factor  must  be 
taken  into  account ;  this  factor  is  the  date  line. 


LONGITUDE  AND  TIME  343 

Suppose  that  a  man  should  start  from  your  schoolhouse 
at  noon  on  Wednesday  and  should  travel  due  west  with 
the  same  rapidity  that  the  earth  turns  on  its  axis,  or  with 
the  same  rapidity  that  the  sun  appears  to  travel  west.  The 
sun  would  then  be  continually  on  the  meridian  of  the  trav- 
eler, hence  for  him  the  time  would  continually  be  Wednes- 
day noon.  He  would  make  the  trip  around  the  earth  in 
twenty-four  hours,  and  when  he  reached  the  starting  point 
the  time  to  him  would  still  be  Wednesday  noon,  since  the 
sun  was  continually  on  his  meridian.  To  those  who  did 
not  make  the  journey  the  time  would  be  Thursday  noon, 
since  twenty-four  hours  elapsed  and  a  night  intervened. 
In  order  to  make  his  reckoning  of  time  correct  the  traveler 
would  have  to  add  one  day.  If  we  suppose  the  traveler 
to  start  at  noon  and  travel  east  with  the  same  rapidity  as 
the  earth  rotates,  or  with  the  same  rapidity  that  the  sun 
appears  to  travel  west,  he  would  arrive  at  his  starting  point 
Friday  noon,  according  to  his  reckoning,  but  Thursday  noon 
according  to  the  reckoning  of  those  who  did  not  make  the 
journey.  In  order  to  rectify  his  dates  the  traveler  must 
drop  a  day. 

In  the  first  illustration  the  traveler  crossed  the  date  line 
from  the  east  towards  the  west;  he  had  to  add  a  day. 
In  the  second  case  the  date  line  was  crossed  from  the  west 
towards  the  east;  the  traveler  had  to  subtract  a  day.  The 
same  adjustment  of  dates  would  need  to  be  made,  no  matter 
what  the  rates  of  travel  might  have  been.  The  supposed 
rates  make  the  illustration  easy  to  follow. 

The  teacher  should  show  the  pupils  that  the  dates  of  the 
traveler  would  have  been  correct  had  he  added  (or  sub- 
tracted, as  the  case  may  be)  a  day  at  any  point  on  his 
journey.  His  date  upon  arrival  at  the  starting  point  would 
have  been  correct  if  he  had  added  a  day  just  after  leaving 
the  school  or  just  before  reaching  it,  or  at  any  intermediate 


344  HOW  TO  TEACH  AEITHMETIC 

point.  This  consideration  leads  to  a  discussion  as  to  why 
the  international  date  line  has  its  present  location.  The 
teacher  will  find  this  point  explained  in  any  good  mathe- 
matical geography.  Space  does  not  permit  a  fuller  dis- 
cussion of  the  question  here  than  to  state  that  most  of  the 
islands  whose  earliest  European  inhabitants  came  by  the 
way  of  the  Cape  of  Good  Hope  have  the  Asiatic  date,  while 
those  that  were  approached  by  way  of  Cape  Horn  have  the 
American  date.  This  accounts  for  the  irregularity  of  the 
date  line. 


CHAPTER  XXV 

LITERAL  ARITHMETIC  AND  ALGEBRA 

There  is  a  marked  tendency  in  this  country  towards  the 
introduction  of  some  form  of  literal  arithmetic  or  of 
algebra  in  the  last  two  years  of  the  grammar  grades.  In 
some  schools  the  work  in  arithmetic  is  finished  by  the  end 
of  the  seventh  grade  or  by  the  end  of  the  first  half  of  the 
eighth,  and  the  remaining  time  is  devoted  largely  to 
algebra.  In  other  schools  some  work  in  algebra  is  intro- 
duced into  the  seventh  or  eighth  grades  without  dropping 
the  arithmetic  and  with  but  little  effort  to  relate  the  two 
subjects  closely.  In  a  few  schools  an  effort  is  made  to 
introduce  literal  arithmetic  or  algebra  as  opportunity  per- 
mits while  arithmetic  is  being  studied,  and  an  effort  is 
made  to  establish  a  vital  connection  between  the  subjects. 
This  last  manner  of  introducing  algebra  is  being  rapidly 
extended  and  adopted.  In  about  35  per  cent  of  the  schools 
of  the  larger  cities  algebra  is  now  taught  in  some  form  in 
the  grades.  Several  arguments  have  been  advanced  to 
justify  this  practice.  It  is  stated  that  a  proper  introduc- 
tion of  the  subject  interests  the  pupils  to  such  an  extent 
that  they  desire  to  enter  the  high  school  in  order  to  extend 
their  knowledge  of  it.  Some  advocate  its  introduction  in 
order  to  make  the  transition  from  grammar  school  to  high 
school  subjects  more  gradual.  Others  introduce  the  simple 
equation,  the  graph,  and  algebraic  formulas  into  the  work 
in  arithmetic  in"  order  to  enrich  the  course  and  to  simplify 
the  solution  of  some  of  the  more  difficult  problems. 

345 


346  HOW  TO  TEACH  AEITHMETIC 

The  Present  Tendency 

The  present  tendency  is  towards  the  introduction  of 
some  work  in  algebraic  formulas,  simple  equations,  and 
graphs  in  connection  with  the  regular  work  in  arithmetic. 
There  is  an  attempt  to  break  down  the  traditional  demar- 
cations between  arithmetic  and  algebra  and  to  correlate 
the  subjects  more  closely.  Many  pupils  of  the  upper 
grammar  grades  do  not  enter  the  high  school,  and  these 
pupils  should  have  the  opportunity  to  learn  the  meaning 
and  the  use  of  algebraic  formulas  and  graphs,  now  so 
frequently  used  in  trade  journals  and  mechanics'  hand- 
books, and  to  appreciate  the  value  of  the  simple  equation 
as  a  tool  in  the  solution  of  problems. 

One  of  the  chief  objections  that  has  been  urged  against 
algebra,  as  it  is  usually  taught  in  the  grammar  schools,  is 
that  it  is  an  attempt  to  complete  a  definite  portion  of  the 
work  of  the  first  year  of  the  high  school.  There  is  often 
too  much  emphasis  upon  definitions,  theory,  and  compli- 
cated processes.  A  pupil  who  does  not  enter  the  high 
school  derives  but  little  benefit  from  a  formal  study  of  the 
four  fundamental  operations  and  factoring  in  algebra. 

In  the  latter  part  of  the  sixth  and  in  the  seventh  and 
eighth  grades  the  teacher  should  not  hesitate  to  introduce 
letters  to  represent  numbers.  It  is  no  more  unnatural, 
after  a  short  time,  for  the  pupil  to  use  ' '  C "  to  represent 
the  cost,  "I"  to  represent  interest,  "V"  to  represent  vol- 
ume, and  "N"  to  represent  a  number  than  for  him  to  use 
N. Y.  to  represent  New  York  or  A.M.  to  indicate  time 
before  noon.  Pupils  soon  appreciate  the  fact  that  calcu- 
lations involving  letters  are  often  simpler  than  those 
involving  figures.  As  Young  says,  "The  use  of  letters 
to  represent  numbers  simplifies  the  treatment  of  some 
types  of  problems  which  otherwise  tend  to  confuse  by 


LITEEAL  ARITHMETIC  AND  ALGEBEA  347 

mere  verbiage.  "  x  If  only  those  processes  that  are  actually 
needed  in  the  solution  of  the  problems  of  arithmetic  are 
emphasized,  but  little  objection  will  be  offered  in  most 
communities  to  the  introduction  of  such  work  in  the  grades. 
The  transformations  that  are  necessary  in  the  solution  of 
simple  equations  and  in  the  evaluation  of  simple  algebraic 
formulas  can  be  readily  understood  by  pupils  of  the  seventh 
and  eighth  grades. 

Strong  protests  are  always  made  when  any  innovations 
in  subject-matter  or  symbols  are  suggested.  Improvements 
are  usually  made  in  spite  of  opposition  and  protest.  Smith 
has  pointed  out  the  fact  that  when  it  was  proposed  to  write 
"4x5"  instead  of  "4  times  5"  strong  protests  were  made 
on  the  ground  that  "4x5"  was  more  abstract  than  "4 
times  5"  and  that  it  would  be  best  to  let  well  enough 
alone.2  The  pupil  should  recognize  the  symbols  of  algebra 
as  a  shorthand  method  of  indicating  magnitudes.  Arith- 
metic does  not  become  algebra  by  the  mere  use  of  letters 
instead  of  numbers.  The  solution  of  many  of  the  problems 
of  common  and  decimal  fractions,  of  ratio  and  propor- 
tion, percentage  and  mensuration,  is  greatly  simplified  and 
abridged  by  the  use  of  letters  for  the  magnitudes  which 
they  represent. 

The  pupil  who  understands  the  following  forms  is  pre- 
pared to  understand  the  fundamental  principles  of  simple 
equations  : 


12-=-?=   6 

7x4  =  1-2 
?-f4  =   8 


instead  of  the  interrogation  point  the  symbol  "n"  is 


1  Young,  ' '  The  Teaching  of  Mathematics, M  p.  243. 

2  Smith,  "The  Teaching  of  Arithmetic/'  pp.  72-73. 


348  HOW  TO  TEACH  ARITHMETIC 

used  to  represent  the  required  number,  its  value  may  be 
readily  found.  If  teachers  will  refrain  from  the  introduc- 
tion of  unnecessary  definitions  and  emphasis  upon  tech- 
nical and  non-essential  points,  they  will  find  that  pupils 
like  to  use  the  literal  notation.  In  this,  as  in  other  sub- 
jects, teachers  often  befog  and  obscure  what  is  otherwise 
clear  to  the  pupil  by  attempting  to  give  minute  explana- 
tions. 

The  pupil  should  appreciate  the  fact  that  an  algebraic 
formula  is  always  an  abridged  and  concise  method  of 
stating  a  fact,  and  he  should  be  able  to  translate  a  formula 
into  words  and  to  translate  a  simple  statement  of  condi- 
tion into  a  formula,  If  "1"  represents  the  length  of  a 
field  in  rods  and  "b"  represents  its  breadth  in  rods,  what 
does  lb-1600  state?  If  "v"  represents  the  volume  of  a 
sphere  and  "r"  represents  its  radius,  what  does  v=|7rr3 
state?  Such  expressions  as  those  above  should  be  as  intel- 
ligible to  the  pupil  when  stated  by  the  use  of  letters  as 
when  stated  in  words.  The  literal  statement  of  a  problem 
is  more  compact  than  the  statement  in  words,  and  the  rela- 
tions between  the  various  factors  involved  may  usually  be 
more  clearly  seen. 

The  value  of  the  equation  as  a  tool  in  the  solution  of 
problems  may  be  impressed  upon  the  pupil  by  requiring 
him  to  solve  some  of  the  problems  of  the  text  without  the 
use  of  letters,  and  then  comparing  with  these  the  solutions 
of  the  same  problems  in  which  literal  notation  and  the 
simple  equation  have  been  employed.  Such  comparisons 
are  often  a  revelation  to  a  pupil. 


CHAPTEE  XXVI 
PRESENT  TENDENCIES  IN  ARITHMETIC 

A  few  decades  ago  the  teachers  of  arithmetic  made  little 
effort  to  appeal  to  the  pupil's  interests  by  utilizing  his 
experiences.  Memory  was  an  important  factor  in  most 
school  work,  and  drill  was  an  essential  part  of  class  instruc- 
tion. To-day  we  recognize  that  the  value  of  a  study  to  a 
pupil  depends  largely  upon  the  amount  of  mental  energy 
that  the  pupil  puts  into  it,  and  that  this  in  turn  is  largely 
dependent  upon  the  interest  with  which  the  pupil  pursues 
the  study.  Interest  in  a  subject  lies  very  near  the  basis 
for  success  in  the  subject.  That  a  child  learns  through 
his  experiences  is  one  of  the  central  facts  of  modern  peda- 
gogy ;  and  as  this  fact  meets  with  more  general  acceptance, 
increasing  emphasis  will  be  placed  upon  the  child's  own 
activities. 

The  pupil  of  the  lower  elementary  school  is  more  or  less 
a  creature  of  the  present.  His  dominant  interests  are  in 
things  that  appeal  to  him  because  of  immediate  utility  or 
pleasure.  The  'strongest  motives  for  good  work  in  the 
lower  grades  are  based  upon  the  pupil's  dominant  interest 
at  the  time.  As  the  pupil  matures  and  his  educational 
horizon  broadens,  his  interest  may  be  aroused  by  the  use 
of  motives  more  or  less  remote  in  time.  It  is  the  duty  of 
the  teacher  to  arouse  interest  in  the  subject  and  to  utilize 
this  interest  to  secure  that  careful  and  consistent  study 
which  is  a  prerequisite  of  the  best  educational  results. 
Persistent  application  is  the  price  that  must  be  paid  for 
successful  mastery  of  any  subject. 

349 


350  HOW  T0  TEACH  ARITHMETIC 

The  School  as  a  Social  Institution 

The  school  is  gaining  recognition  as  a  social  institution 
and  we  are  beginning  to  realize  that  social  efficiency  means 
other  than  mere  business  efficiency.  The  pupil  has  the 
right  to  be  informed  in  regard  to  the  broader  aspects  of 
modern  social,  industrial,  and  commercial  Activities,  and 
it  is  the  duty  of  the  school  to  see  that  he  acquires  this 
information.  In  so  far  as  this  information  involves  the 
larger  quantitative  aspects  of  those  activities  it  may  prop- 
erly be  included  in  a  course  in  arithmetic.  "Mind  fur- 
nishing and  mind  training  go  hand  in  hand." 

Problem  Material 

One  of  the  marked  features  of  the  arithmetic  of  to-day 
is  the  attempt  to  adapt  the  problem  material  to  the  needs 
and  interests  of  the  pupil  instead  of  adapting  the  pupil  to 
the  problem,  as  was  frequently  attempted  by  the  older 
texts.  It  must  be  admitted  that  the  result  is  often  a 
loss  to  the  pupil.  Numerous  problems  relating  to  the  com- 
mon phases  of  community  life  are  being  introduced.  The 
aim  %  is  to  secure  the  maximum  amount  of  self -activity  on 
the  part  of  the  pupil  by  confronting  him  with  problems 
which  appeal  to  him  as  concrete  and  vital.  There  is  a 
growing  recognition  of  the  fact  that  there  should  be  a 
legitimate  motive  or  purpose  underlying  a  problem,  and 
that  as  many  problems  as  possible  should  be  more  or  less 
related  to  matters  that  are  within  the  experience  of  the 
pupil.  A  problem  that  appeals  to  an  adult  as  real  and 
vital  may  not  make  the  same  appeal  to  a  child.  A  prob- 
lem that  appeals  strongly  to  a  pupil  of  the  fifth  or 
sixth  grade  .may  be  of  little  value  to  the  pupil  of  the 
second  or  third  grade,  even  with  smaller  numbers;  and 
some  of  the  number  games  of  the  lower  grade  would  be 


PRESENT  TENDENCIES  IN  ARITHMETIC  351 

out  of  place  in  the  higher  grades.  The  appeal  should  be 
to  the  interests  and  activities  of  the  pupil  in  and  out  of 
school,  and  the  interests  of  the  adult  should  be  regarded 
as  of  subordinate  importance.  A  problem  is  not  of  maxi- 
mum value  merely  because  it  is  about  a  factory,  a  store, 
or  a  bank.  It  should  be  of  a  type  that  is  actually  met  by 
those  who  do  the  world's  work,  and  the  data  involved 
should  be  within  the  intelligence  and  the  experience  of 
the  pupil.  A  problem  may  be  concrete  and  full  of  sig- 
nificance to  one  pupil  and  not  at  all  so  to  another  pupil. 
Myers  has  pointed  out  the  fact  that  "children's  problems 
are  not  merely  men's  or  women's  problems  cracked  up  into 
smaller  bits.  They  must  differ  qualitatively  as  well  as 
quantitatively."1  Most  pupils  are  anxious  to  solve  prob- 
lems that  actually  come  within  their  own  experience.  The 
fact  that  a  pupil  is  interested  to  know  how  to  solve  a 
problem  does  not  necessarily  imply  that  he  will  be  able 
to  solve  it,  but  much  has  been  gained  when  problems  have 
been  so  chosen  that  pupils  are  willing  and  eager  to  learn 
how  to  solve  them.  Interest  begets  effort,  and  effort  prop- 
erly directed  produces  results.  Any  problem  that  appeals 
to  the  pupil  may  legitimately  be  used  in  arithmetic,  pro- 
vided it  does  not  give  a  false  idea  of  social,  industrial,  and 
commercial  activities  of  the  day.  Text-books  suggest  nu- 
merous types  of  problems,  and  the  teacher  should  supple- 
ment them  by  problems  of  a  local  character.  Pupils  should 
be  encouraged  to  bring  in  problems  that  appeal  to  them  as 
interesting. 

Many  texts  of  former  years  went  to  the  extreme  of  devot- 
ing too  little  attention  to  the  applications  of  arithmetic, 
and  the  reaction  against  this  tendency  is  likely  to  carry 
us  too  far  in  the  other  direction.  Many  problems  dealing 
with  the  various  relations  and  properties  of  numbers  are 

i '  *  Arithmetic  in  Public  Education, ' 9  Myers,  pp.  8-9. 


352  HOW  TO  TEACH  AEITHMETIG 

more  interesting  to  the  pupil  than  certain  problems  of  the 
shop.  The  essential  thing  is  that  the  problem  shall  really 
appeal  to  the  pupil  as  interesting.  To  use  only  those  prob- 
lems that  involve  buying  and  selling,  measuring  and  esti- 
mating, and  the  various  business  procedures,  is  to  eliminate 
from  arithmetic  many  problems  that  are  intensely  inter- 
esting to  the  pupils.  It  is  as  unwise  to  require  pupils  to 
solve  numerous  problems  involving  applications  of  which 
they  have  no  conception  as  to  eliminate  all  of  the  interest- 
ing problems  of  former  years  merely  because  they  find  no 
immediate  application  in  present-day  activities.  Those  who 
have  adopted  the  extreme  utilitarian  view  and  insist  that 
nothing  be  taught  except  that  which  "functions  in  the 
immediate  present"  are  not  numerous,  but  their  influence 
is  quite  widely  felt.  A  pupil  may  become  intensely  inter- 
ested in  solving  a  problem  which  appeals  to  his  fancy, 
even  though  it  is  in  no  way  related  to  the  so-called 
practical.  In  the  broader  sense  of  the  term,  any  problem 
in  which  the  pupil  is  really  interested,  and  which  does  not 
give  him  false  ideas,  is  practical  for  him,  because  it  helps 
to  develop  his  ability  to  understand  number  relations. 

Some  enthusiasts  have  introduced  into  the  seventh  and 
eighth  grades  numerous  problems  that  are  social  in  con- 
tent but  involve  nothing  but  the  mathematics  of  the  pri- 
mary school.  The  introduction  of  problems  of  this  type 
may  be  quite  valuable  in  classes  in  elementary  sociology 
and  economics,  but  such  problems  do  not  contribute  to 
mathematical  insight  or  skill. 

We  must  try  to  make  arithmetic  interesting  to  the  pupils, 
and  a  judicious  selection  of  problems  is  one  means  of 
accomplishing  this  end.  However,  practical  utility  is  not 
the  only  criterion  by  which  the  worth  of  a  problem  should 
be  judged.  One  of  the  keenest  pleasures  a  pupil  expe- 
riences in  the  study  of  arithmetic  comes  from  the  ability 


PRESENT  TENDENCIES  IN  ARITHMETIC  353 

to  overcome  the  difficulties  involved  in  the  solution  of 
problems.  The  joy  of  conquest  is  a  large  factor  in  the 
pupil's  intellectual  life.  Too  often  pupils  experience  little 
of  the  pleasure  that  comes  from  the  successful  mastery  of 
a  difficult  task.  The  joy  of  intellectual  effort  and  of  mental 
acquisition  are  too  little  known  by  the  modern  pupil. 
Patience  and  persistency  of  effort  are  not  sufficiently 
emphasized. 

It  is  difficult  to  secure  an  adequate  supply  of  practica\ 
problems  involving  the  mathematical  principles  which 
must  be  mastered,  but  great  advance  has  been  made  in 
recent  years  in  this  respect,  and  no  doubt  further  search 
will  reveal  other  types  of  problems  of  the  desired  kind. 
At  present,  however,  a  pupil  cannot  acquire  the  desired 
facility  and  accuracy  in  mathematical  operations  and  de- 
velop sufficiently  his  power  .to  discriminate  the  essential 
relations  of  the  various  elements  involved  if  the  problems 
are  restricted  to  the  utilitarian  types  that  are  immediately 
within  his  experience.  Furthermore,  it  is  not  true  that  the 
pupil's  only  interest  is  in  this  type  of  problem.  He  may 
become  interested  in  the  solution  of  a  problem  that  has 
no  immediate  relation  to  utilitarian  ends ;  one  that  cannot 
be  directly  or  indirectly  correlated  with  the  practical.  If, 
however,  the  problem  makes  a  strong  appeal  to  him;  if 
it  engages  his  attention  and  challenges  his  powers,  it  is 
serving  the  purpose.  A  problem  that  appeals  to  the  pupil 
"just  for  the  joy  of  the  doing"  must  not  be  left  out,  even 
though4  it  has  no  other  value.  Such  problems  serve  a 
useful  purpose  and  are  too  few. 

Not  every  task  in  life  or  in  school  is  a  pleasant  one  at 
the  time  it  is  being  performed.  We  may  wish  that  it 
were  so  and  may  strive  to  make  it  so,  but  the  fact  remains 
that  some  things  must  be  done  both  in  and  out  of  school 
that  are  not  in  themselves  interesting.  Such  work  may 


354  HOW  TO  TEACH  AEITHMETIG 

have  a  very  direct  bearing  on  other  work  that  will  be 
interesting  because  of  the  mastery  of  something  in  itself 
uninteresting.  The  thoughtful  teacher  welcomes  every 
suggestion  that  makes  any  school  activity  more  interesting 
and  is  continually  seeking  for  ways  and  means  to  accom- 
plish this  end,  but  no  way  has  yet  been  discovered  to  make 
all  tasks  interesting  to  all  the  pupils  all  the  time.  Our 
pupils  need  to  have  developed  within  them  the  habit  of 
sticking  to  a  task  until  it  is  successfully  completed,  even 
though  the  doing  of  the  work  may  not  appeal  to  them  as 
interesting.  No  good  teacher  will  go  to  the  extreme  of 
giving  pupils  an  uninteresting  task  to  perform  merely 
because  it  is  uninteresting,  but  it  is  almost  as  great  an 
error  to  eliminate  everything  that  does  not  immediately 
engage  the  pupil's  interest.  The  teacher  who  is  interested 
in  his  work  usually  finds  that4 his  enthusiasm  in  the  subject 
is  contagious  among  his  pupils.  If  the  teacher  increases 
his  margin  of  knowledge  and  becomes  a  master  of  the  field 
that  he  is  teaching,  he  usually  finds  that  interest  begets 
interest,  and  that  the  more  he  puts  into  his  subject,  the 
more  response  he  gets  from  his  pupils  and  the  more  interest 
is  aroused.  Much  grind  is  due  to  the  fact  that  the  teacher 
has  neither  the  ability  nor  the  energy  to  enliven  the  subject 
or  to  awaken  interest  and  enthusiasm  by  being  interested 
and  enthusiastic  himself. 

Omission  and  Introduction 

The  tendency  to-day  is  to  omit  from  arithmetic  any 
topic  that  time  or  changing  social  conditions  has  rendered 
obsolete  or  purely  technical  for  a  small  group,  and  to 
revitalize  the  topics  that  remain  so  that  they  will  repre- 
sent actual  conditions  of  the  present.  There  is  increasing 
emphasis  upon  the  fact  that  school  work  in  arithmetic 
should  prepare  the  pupil  to  deal  with  "out  of  school 


PRESENT  TENDENCIES  IN  ARITHMETIC  355 

problems."  Arithmetic  should  train  the  pupil  to  see  the 
world  from  a  quantitative  point  of  view.  Emphasis  is 
placed  upon  the  types  of  problems  that  frequently  occur 
in  the  larger  "out  of  school"  activities,  and  no  attempt  is 
made  to  present  every  kind  and  form  of  problem  that  may 
be  practical  in  minor  trades  in  which  the  pupils  have  no 
immediate  concern.  The  school  must  give  the  pupil  a 
quantitative  knowledge  of  those  facts  which  he  must  some 
_  day  know  in  order  to  assume  his  place  in  social  and  indus- 
trial life.  Problems  of  food  and  clothing  supply,  of  trans- 
portation, of  buying  and  selling,  of  building,  of  mining,  of 
city  and  county  administration  are  emphasized. 

Problems  of  the  farm  are  receiving  greater  emphasis  in 
rural  communities,  and  problems  involving  the  numerous 
activities  of  city  life  are  emphasized  in  urban  communities. 
Well-executed  and  accurate  pictures  which  really  aid  in 
the  understanding  of  number  relations  are  being  used 
increasingly  in  text-books. 

Unduly  long  and  complicated  problems  are  being  omit- 
ted and  greater  emphasis  is  being  placed  upon  the  essen- 
tials of  arithmetic.  "There  is  a  growing  belief  that 
the  aim  of  the  work  in  arithmetic  should  be  limited  to 
accuracy  and  a  reasonable  facility  in  the  fundamental 
operations — addition,  subtraction,  multiplication,  and  divi- 
sion of  whole  numbers  and  simple  fractions ;  and  to  simple 
practical  problems  involving  the  operations  together  with 
some  instruction  in  percentage  and  its  simplest  applica- 
tions to  interest,  trade  discount,  taxes,  and  insurance."1 

"The  obsolete  and  the  relatively  infrequent,  the  over- 
complex  and  the  wasteful  processes  of  the  old  arithmetic 
tend  to  disappear." 

1  Paul  H.  Hanus,  Harvard  College,  in  a  Report  on  the  Program  of 
Studies,  1911. 

2  Suzzalo,  ' '  The  Teaching  of  Primary  Arithmetic. ' ' 


356  HOW  TO  TEACH  AEITHMETIC 

The  time  saved  by  the  omission  of  obsolete  topics  and 
the  abridged  treatment  of  some  others  is  being  utilized  in 
a  number  of  ways.  We  are  beginning  to  appreciate  the 
extent  to  which  arithmetic  may  be  made  to  contribute  to 
social  insight.  Pupils  are  shown  how  the  world  uses  its 
mathematics  by  visits  to  centers  of  commercial  and  indus- 
trial activity  and  the  setting  of  the  problem  and  its  solu- 
tion are  studied.  Instead  of  devoting  all  of  the  time  to 
the  solution  of  problems  about  a  factory  or  a  bank,  the 
modern  class  occasionally  visits  these  institutions.  A  good 
course  in  arithmetic  to-day  includes  a  consideration  of  the 
following  topics,  all  of  which  have  been  introduced  re- 
cently :  The  saving  and  loaning  of  money ;  the  investing 
of  money ;  modern  banking  methods ;  keeping  of  simple 
accounts ;  a  study  of  tax  levies ;  public  expenditures  and 
insurance  from  the  social  point  cf  view.  Much  of  the 
work  in  these  topics  must  be  of  the  informational  charac- 
ter, and  belongs  quite  as  much  to  a  course  in  civics  and 
economics  as  to  arithmetic. 

The  various  factors  which  make  for  the  socialization  of 
arithmetic  will,  if  properly  controlled  and  judiciously  used, 
increase  the  pupil's  efficiency  in  the  subject.  .  Many  teach- 
ers will  carry  the  idea  to  absurd  extremes,  and  as  a  result 
their  pupils  will  have  but  little  mastery  of  the  essential 
number  facts.  Any  good  tendency  may  become  bad  when 
carried  too  far. 

Skill  in  Computation 

The  arithmetic  of  to-day  is  generally  less  abstract  and 
formal  than  that  of  a  few  decades  ago.  Manual  training, 
industrial  and  household  arts,  geography,  history,  agri- 
culture, and  elementary  physics  are  being  utilized  as 
sources  of  problems.  Arithmetic  is  less  a  series  of  abstract 
exercises  and  more  a  tool  useful  in  other  subjects.  The 


PRESENT  TENDENCIES  IN  ARITHMETIC  357 

fact  that  much  of  the  work  is  being  motived  does  not  mean 
that  the  drill,  so  necessary  to  fix  in  mind  the  fundamental 
number  facts,  should  be  relegated  to  a  subordinate  posi- 
tion. The  more  arithmetic  is  used  in  practical  ways,  the 
more  must  the  pupils  realize  the  need  of  mastery  of  the 
fundamental  number  relations;  and  this  in  itself  is  one 
motive  for  drill.  There  is  increasing  emphasis  to-day  on 
systematic  drill.1  There  is  a  growing  belief  in  the  neces- 
sity of  concentrating  the  attention  at  times  upon  the 
operations  alone.  If  our  pupils  are  to  learn  to  compute 
with  facility  and  accuracy,  they  must  have  a  great  deal  of 
practice  in  computing.  Arithmetic  is  both  a  science  and 
an  art.  Exercises  are  necessary  to  develop  skill. 

The  Solution  of  Problems 

When  a  pupil  is  required  to  solve  a  problem,  he  must 
first  comprehend  the  situation  presented  and  must  then 
decide  upon  the  operations  to  be  used,  then  the  actual  com- 
putations must  be  made.  A  large  number  of  exam- 
ples must  be  given  in  order  to  develop  skill  in  compu- 
tation. By  the  time  the  pupil  has  reached  the  latter  grades 
of  the  grammar  school  the  fundamental  operations  ought 
to  be  so  thoroughly  mastered  that  his  mind  is  free  to 
devote  itself  to  the  solution  of  the  problem  rather  than 
to  the  performance  of  the  operation.  The  solution  of  a 
problem  involves  knowing  what  to  do,  and  the  doing  of  it 
should  be  a  minor  factor  in  the  upper  grades.  Pupils 
need  a  great  deal  of  drill  in  interpreting  problems;  this 
phase  of  the  work  has  received  too  little  emphasis  in  most 
schools.  It  is  wise  to  require  a  pupil  frequently  to  trans- 
form a  "problem"  into  an  "example";  that  is,  to  require 
him  to  analyze  the  data  given  in  the  problem,  in  order  to 

i  See  chapter  on  Drill. 


358  HOW  TO  TEACH  AEITHMETIC 

determine  the  proper  processes  to  be  used,  writing  down  in 
one  column  the  analysis  in  systematic  form,  and  in  another 
column  doing  the  necessary  computing,  using  all  the  short 
cuts  possible.  In  the  one  .column,  emphasis  is  put  upon 
accuracy  of  thought  processes,  in  the  other,  on  accuracy 
of  computation.  The  modern  course  of  study  limits  the 
problems  largely  to  matter  coming  within  the  pupil's 
experience,  and  the  modern  school  patron  is  demanding 
efficiency  within  this  narrowed  field.  The  solution  of  a 
large  number  of  problems  of  medium  difficulty  is  of  more 
value  to  the  pupil  than  the  solution"  of  a  few  problems  of 
great  difficulty.  The  teacher  of  to-day  is  placing  emphasis 
upon  oral  arithmetic,  and  we  are  returning  to  a  reasonable 
use  of  that  important  phase  of  the  work. 

The  Analysis  of  Problems 

Not  only  is  there  a  tendency  to  master  the  quantitative 
side  of  life  by  utilizing  the  pupil's  experience  and  by  cor- 
relating the  work  in  arithmetic  with  the  work  in  other 
subjects,  but  there  is  a  decided  tendency  to  foster  a  spirit 
of  inquiry  and  to  develop  the  power  to  interpret  number 
relations.  Instead  of  requiring  a  pupil  to  solve  a  problem 
according  to  some  type  form,  we  seek  to  encourage  his 
originality  and  individuality  by  permitting  more  flex- 
ibility in  analysis  than  was  formerly  allowed.  We  en- 
courage him  to  choose  the  method  that  seems  best  to  him, 
and  then  we  ask  him,  at  times,  to  justify  his  choice.  The 
rigidity  of  full  logical  form  in  analysis  is  not  considered 
so  important  as  in  prevo'us  years.  A  pupil  who  always 
associates  a  particular  process  with  specific  words  of  rela- 
tion used  in  a  problem  is  often  unable  to  solve  the  problem 
when  the  phraseology  is  even  slightly  changed.  Marked 
variation  in  the  difficulty  of  a  problem  may  be  caused  by  a 


PRESENT  TENDENCIES  IN  ARITHMETIC  359 

difference  of  phraseology.  By  a  judicious  lack  of  uniform- 
ity in  the  phraseology  we  force  the  pupil  to  rely  upon  his 
own  resources  in  solving  them.  The  pupil  is  encouraged 
to  see  the  relation  between  what  is  given  and  what  is 
required,  and  to  choose  the  appropriate  process.  Super- 
ficial and  thoughtless  associations  are  discouraged. 

The  difficulty  in  a  given  problem  usually  lies  in  the  fact 
that  the  pupil  is  not  able  to  break  the  problem  up  into  its 
various  steps  and  to  apprehend  the  order  of  the  steps. 
Unless  he  has  more  or  less  ability  to  analyze  problems,  his 
knowledge  of  arithmetic  will  be  more  or  less  mechanical. 
The  pupil  who  continually  depends  upon  rules  or  type 
problems  has  attained  but  little  mastery  of  the  subject. 
He  should  be  encouraged  to  devise  solutions  of  his  own. 
The  more  insight  he  has  into  the  relationships  involved  in 
a  given  problem,  the  greater  freedom  will  he  have  from 
mechanical  procedure. 

Rationalizing  the  Processes 

There  is  a  marked  tendency  to-day  to  rationalize  the 
processes  in  arithmetic.1  It  is  not  assumed  that  the  pupil 
will  be  able  to  give  a  clear-cut  and  logical  explanation  of 
every  process,  but  each  process  should  be  developed  and 
explained  in  such  a  way  that  the  entire  procedure  appeals 
to  him  as  reasonable.  For  example,  instead  of  being  told 
to  invert  the  divisor  in  division  of  fractions  and  multiply 
by  the  dividend,  he  is  first  shown  why  such  a  procedure 
will  give  the  correct  result. 

Unity  of  Arithmetic 

i 

Another  tendency  in  the  teaching  of  arithmetic  is  the 
effort  to  develop  the  subject  as  a  whole  rather  than  as  a 

i  See  chapter  on  Rules  and  Analysis. 


360  HOW  TO  TEACH  AEITHMETIC 

series  of  more  or  less  unrelated  topics.  More  emphasis  is 
being  placed  to-day  on  the  mastery  of  principles  and  less 
on  the  mastery  of  rules  and  definitions  than  was  the  case 
in  former  years.  The  pupil  who  fails  to  grasp  the  few 
simple  underlying  principles  of  the  subject  has  seen  but 
little  of  the  beauty  and  the  simplicity  of  arithmetic.  The 
number  of  distinct  mathematical  principles  involved  is  sur- 
prisingly small,  but  unless  a  pupil  grasps  these  principles 
the  whole  subject  lacks  organic  unity  in  his  mind.  It  is 
important  to  analyze  and  to  classify  topics;  to  be  able  to 
distinguish  points  of  similarity  and  of  difference;  to  de- 
velop the  power  to  see  relations  between  topics  that  are 
based  on  the  same  underlying  principles.  As  McMurry 
says,  "Merely  to  go  through  a  text -book  without  picking 
up  the  strings  and  tying  them  together  is  to  fail  in  a  most 
essential  thing.'71  The  pupil  who  does  not  see  the  unity 
in  arithmetic  fancies  that  he  is  dealing  with  something 
entirely  new  from  the  mathematical  point  of  view  every 
time  he  takes  up  a  new  topic,  whereas  he  may  be  dealing 
with  a  new  phase  of  a  topic  long  familiar  to  him.  Keal 
knowledge  implies  organization,  and  organization,  in  turn, 
implies  more  or  less  unity.  Spencer  says,  "When  a  man's 
knowledge  is  not  in  order,  the  more  of  it  he  has  the  greater 
will  be  his  confusion  of  thought.  When  the  facts  are  not 
organized  into  faculty,  the  greater  the  mass  of  them  the 
more  will  the  mind  struggle  under  its  burden,  hampered 
instead  of  helped  by  its  acquisition." 

There  is  a  tendency  to-day  not  only  to  unify  the  sub- 
ject as  a  whole,  but  to  introduce  more  or  less  unity  into 
the  assignment  of  problems  for  a  lesson  or  a  group  of 
lessons.  The  custom  has  been  to  group  problems  more  or 
less  into  series  involving  the  same  processes  but  different 
in  all  other  respects.  The  present  tendency  is  to  "attain  a 

i  See  McMurry,  i '  How  to  Study, ' '  p.  79. 


PRESENT  TENDENCIES  IN  ARITHMETIC 

~T 

& 

more  approximate  unity  within  the  subject-matter  of  the 
problems  themselves." 

Formal  Definitions 

There  is  a  marked  tendency  to-day  to  minimize  the 
importance  of  formal  definitions,  especially  in  the  lower 
grades.  The  meaning  of  each  term  used  should  be  clearly 
understood,  but  it  is  not  wise  to  require  formal  definitions 
in  arithmetic  from  young  children.  A  pupil  in  the  lower 
grades,  if  required  to  give  a  formal  definition  of  an  arith- 
metical term,  usually  repeats  the  exact  words  of  the  text- 
book, and  the  real  meaning  is  but  little  understood.  Too 
often  the  pupil  who  states  a  definition  most  glibly  is 
repeating  words  without  expressing  ideas.  It  is  probable 
that  no  child  gets  the  idea  of  number  from  a  definition 
of  it.  In  the  later  grammar  grades  some  attention  may 
properly  be  given  to  the  definition  of  terms,  but  no  attempt 
should  be  made  to  define  a  term  until  it  has  been  illus- 
trated. Pupils  should  be  trained  to  formulate,  the  few 
definitions  required,  in  a  clear  and  concise  manner,  and  the 
teacher  should  determine  by  repeated  illustrations  whether 
th*e  definition  has  the  proper  content  in  the  pupil's  mind. 
The  few  definitions  that  are  learned  should  be  expressed 
in  clear  and  forceful  language.  The  attempt  to-day  in 
arithmetic  is  to  teach  the  pupils  "to  observe,  to  think,  to 
lo,  rather  than  to  repeat  and  to  memorize."1 

Scientific  vs.  Empirical  Basis 

Another  tendency  in  arithmetic  is  the  attempt  to  estab- 
lish the  various  methods  of  presenting  the  subject  upon  a 
scientific  rather  than  an  empirical  basis.  Serious  attempts 

i  Howland. 


302 


HOW  TO  TEACH  ARITHMETIC 


are  being  made  to  determine  by  means  of  controlled  experi- 
ments the  relative  value  of  the  methods  that  are  generally 
advocated,  and  to  substitute  verified  conclusions  for  indi- 
vidual opinion.  In  these  days  of  educational  unrest  and 
reform  it  is  not  strange  that  advocates  of  extreme  methods 
should  appear.  Not  every  suggestion  for  reform  is  in  the 
right  direction,  and  it  is  as  necessary  to  reject  new  errors 
as  it  is  to  eliminate  old  ones.  "We  must  not  lose  our 
bearings  in  the  midst  of  the  unscientific  radicalism  of  the 
day/'  We  must  examine  with  great  care  that  which  the 
pedagogic  alchemist  would  have  us  try.  We  must  follow 
the  conclusion  of  our  best  thoughts  and  not  be  bound  by 
the  prejudice  of  years  of  practice.  The  basis  for  any 
successful  system  of  pedagogy  must  be  laid  in  the  expe- 
rience of  preceding  generations.  It  is  necessary  to  master 
the  pedagogy  of  the  mathematics  of  the  past,  its  aims,  its 
methods,  its  devices,  in  order  to  estimate  the  advantages 
of  previous  attainments  and  to  adapt  the  best  to  the  needs 
of  our  generation.  The  teacher  who  knows  the  history  of 
his  subject  is  able  to  make  a  conservative  selection  among 
the  many  radical  suggestions  and  schemes  for  improve- 
ment, each  of  which  probably  has  a  foundation  of  truth 
but  is  given  an  exaggerated  importance  by  its  chief  advo- 
cate. There  must  be  a  careful  weighing  and  adjusting. 
Through  a  knowledge  of  the  history  of  arithmetic  we  gain 
an  ability  to  deal  with  the  modern  suggestions  for  improve- 
ment which  we  shall  be  called  upon  to  judge.  We  also 
avoid  the  ridiculous  situation  of  advocating  as  new  a 
method  or  device  proposed  and  discarded  centuries  ago. 

Investigations  in  Arithmetic 

In  the  chapter  on  Investigations  in  Arithmetic  the  con- 
clusions of  the  noteworthy  scientific  experiments  that  have 
been  made  are  briefly  enumerated.  More  of  our  teaching 


PRESENT  TENDENCIES  IN  ARITHMETIC  363 

is  based  upon  the  results  of  such  investigation  to-day  than 
ever  before  in  the  history  of  education,  and  more  studies 
are  imperatively  needed  in  every  field  of  school  work.  There 
has  been  too  much  tacit  acquiescence  to  general  opinion  in 
education.  Not  every  teacher  has  the  ability  or  the  training 
that  justifies  him  in  attempting  to  plan  and  to  conduct  an 
experiment  to  determine  the  relative  values  of  different 
methods  of  procedure.  Numerous  factors  may  enter  into 
the  investigation,  and  unless  these  are  given  proper  weight 
the  entire  experiment  may  be  of  no  value.  Most  teachers, 
however,  can  cooperate  with  an  investigator  of  ability,  and 
can  thus  aid  in  increasing  the  amount  of  knowledge  based 
on  controlled  experiment. 

Text-books 

Another  tendency  in  the  arithmetic  of  to-day  is  seen  in 
the  organization  of  the  material  of  the  text-book  and  the 
attitude  of  the  teacher  toward  the  texts.  Some  years  ago, 
when  the  faculty  psychology  was  in  vogue,  there  was  ex- 
treme regard  for  logical  arrangement.  The  disciplinary 
side  of  arithmetic  was  emphasized  much  more  than  the 
practical  side.  Gradually  the  demands  for  the  elimina- 
tion of  the  obsolete  and  the  impractical  became  more 
insistent,  and  we  are  witnessing  the  results  of  this  move- 
ment. The  old  arithmetic  sought  to  give  a  complete  treat- 
ment of  each  topic  before  proceeding  to  the  next.  The 
rise  of  the  kindergarten  movement  destroyed  this  ideal, 
for  it  was  soon  recognized  as  impossible  to  give  a  thorough 
treatment  of  any  topic  to  a  child  of  five,  if  his  interests  and 
capacities  were  considered.  The  Grube  method  was  a 
prominent  influence  against  the  extreme  topical  plan,  but 
it  was  just  as  logically  impossible.  The  eagerness  with 
which  the  method  was  adopted  indicated  the  desire  to 


364  HOW  T0  TEACH  AEITHMETIC 

discard  the  extreme  topical  plan.  The  wearisome  monotony 
of  the  Grube  method  and  the  impossible  thoroughness 
which  was  sought  tended  to  destroy  the  child's  interest  in 
number. 

The  Spiral  Plan 

A  direct  outgrowth  of  the  reaction  against  the  extreme 
topical  plan  and  the  Grube  method  was  the  Spiral  Plan. 
It  sought  to  continue  the  Grube  method  in  part,  but  elimi- 
nated the  idea  of  extreme  thoroughness.  The  plan  met 
with  wide  acceptance,  and  numerous  texts  based  on  this 
idea  appeared.  As  more  topics  were  Introduced,  the  coils 
of  the  spiral  shortened  and  the  pupils  were  nauseated  by 
the  frequent  recurrence  of  the  same  topics.  The  reaction 
against  the  extreme  spiral  form  set  in,  and  the  trend  is 
now  towards  much  longer  spirals.  In  many  sections  of  the 
country  there  is  a  very  strong  sentiment  against  any  form 
of  spirals.  The  best  texts  of  to-day  use  neither  the 
extreme  spiral  nor  the  extreme  topical  arrangement  of 
subjects,  but  attempt  to  use  the  best  features  of  each  plan. 
The  ideal  arrangement  is  probably  somewhere  between  the 
two  extremes. 

Influence  of  Text-books  on  Teaching 

In  1895  a  book  entitled  "The  Psychology  of  Number" 
appeared.  The  authors,  McClellan  and  Dewey,  advocated 
the  idea  of  teaching  arithmetic  by  placing  great  emphasis 
on  measurement  as  the  basis  of  number  ideas.  The  book 
exerted  a  marked  influence  on  teachers,  and  many  courses 
were  organized  about  measurement  as  the  basis  of  number 
concepts. . 

Text -books  influence  the  teaching  of  arithmetic  to  a  great 
degree.  Thousands  of  teachers  are  largely  dependent  upon 


PRESENT  TENDENCIES  IN  ARITHMETIC  365 

cext-books  for  their  method,  organization,  material,  and 
sequence  of  topics.  Originality  and  initiative  are  as  rare 
among  teachers  as  among  individuals  in  other  vocations 
and  professions.  Most  teachers  are  followers,  not  leaders. 
There  is  to-day  a  growing  independence  among  teachers  in 
regard  to  text-books.  Many  of  the  older  texts  were  fol- 
lowed slavishly,  and  a  few  of  them  even  indicated  the 
questions  that  the  teacher  was  to  ask  and  specified  the 
pages  on  which  the  answer  was  to  be  found.  There  is 
a  growing  recognition  of  the  fact  that  no  book  can 
adequately  meet  the  needs  of  any  class  at  every  instant. 
It  is  the  duty  of  the  teacher  to  introduce  numerous  exer- 
cises not  in  the  text,  to  give  many  of  the  problems  a  local 
setting,  and  ta  omit  such  topics  and  problems  as  are  not 
adapted  to  the  needs  and  capacities  of  the  pupils.  It  was 
formerly  regarded  as  a  confession  of  weakness  for  a  teacher 
to  omit  any  topic  or  any  problem  found  in  the  text.  We 
are  beginning  to  recognize  that  a  judicious  elimination 
and  supplementing  of  the  material  of  the  text-books  is  an 
indication  of  strength.  The  text-book  should  not  be  the 
teacher's  creed.  Many  teachers  are  prone  to  view  with 
suspicion  any  method  of  procedure  that  a  pupil  ventures 
to  introduce  unless  it  corresponds  with  the  method  of  the 
text.  Such  an  attitude  towards  a  text-book  tends  to  re- 
press rather  than  to  encourage  that  spirit  of  inquiry  and 
of  originality  which  arithmetic  should  seek  to  cultivate. 
The  use  of  a  text-book  is,  in  most  cases,  economical  both 
of  time  and  of  energy.  The  text-book  should  be  the  basis 
for  the  work.  A  good  teacher  will  get  good  results  in 
spite  of  a  poor  text-book,  but  a  well-written  text  increases 
the  efficiency  of  the  teacher.  A  poor  teacher  will  obtain 
poor  results  from  any  text-book. 

Most  authors  of  text-books  are   followers  rather  than 
leaders  in  educational  thought.    If  an  author  is  too  radical 


366  HOW  T0  TEACH  AEITHMETIC 

or  too  conservative  in  his  material  or  method,  the  sale  of 
his  book  is  greatly  curtailed. 

Preparation  of  a  Course  of  Study 

The  recent  tendency  of  many  teachers  to  break  away 
entirely  from  written  texts  and  to  prepare  courses  of  study 
in  arithmetic  which  they  think  are  better  adapted  to  their 
localities,  are  not  to  be  commended.  A  good  course  of 
study  involves  a  large  number  of  factors,  and  the  old 
maxim  that  everybody  knows  better  than  anybody  is  par- 
ticularly applicable  here.  We  should  not  lightly  discard 
the  accumulated  wisdom  of  the  past  and  strive  to  build  up 
a  course  of  study  solely  from  our  own  theories.  Not  every- 
thing that  was  done  in  the  schools  of  other  years  was  bad, 
and  not  everything  was  good,  but  to  discard  both  good  and 
bad  is  folly.  To  prepare  a  good  course  of  study  in  arith- 
metic is  a  very  difficult  task.  It  demands  not  only  a 
knowledge  of  local  conditions  where  the  course  is  to  be 
used,  but  it  demands  a  knowledge  of  the  history  of  the 
subject.  It  demands  a  breadth  of  view  and  a  sanity  of 
judgment  that  not  many  possess.  It  demands  a  knowledge 
of  the  best  that  is  being  done  in  other  countries.  It  de- 
mands clear  ideas  as  to  the  purpose  for  which  arith- 
metic should  be  taught  in  the  schools.  Local  adaptations 
are  comparatively  easy  to  make,  but  a  good  course  of  study 
is  rare  indeed. 


CHAPTER  XXVII 
COURSES   OF   STUDY 

One  of  the  most  satisfactory  studies  of  a  local  situation 
as  a  factor  in  determining  the  character  of  a  course  of 
study  in  arithmetic  that  has  come  to  our  notice  was  made 
by  Mr.  G.  M.  Wilson,  formerly  Superintendent  of  Schools 
at  Connersville,  Indiana.  The  problem  was  attacked 
(1)  through  grade  meetings,  (2)  by  comparing  forty- 
seven  representative  courses  of  study,  and  (3)  by  securing 
the  testimony  and  assistance  of  local  business  men. 

The  course  of  study  resulting  from  this  investigation  is 
organized  upon  the  basis  of  social  utility.  Obsolete  and 
complicated  materials  are  eliminated.  The  arithmetic  of 
new  forms  of  business  practice  are  introduced.  The  sim- 
plest forms  of  solution  are  encouraged. 

The  grade  occurrence  of  arithmetic  topics  as  shown  by 
these  forty-seven  courses  of  study  was : 

GRADES 
Subject  I       II      III      IV      V      VI    VII  VIII 


Numeration    ... 

T7 

37 

33 

18 

7 

4 

A 

1 

Notation   

SI 

39 

33 

18 

7 

4 

4 

1 

Relation  of  Numbers  
Addition   

10 
23 

11 
39 

6 
33 

5 

28 

5 
13 

4 
9 

4 
6 

3 
3 

Subtraction   

?,4 

3'9 

34 

26 

13 

9 

6 

3 

Multiplication        .          .... 

10 

28 

30 

25 

15 

15 

10 

10 

Division   

2 

16 

21 

34 

27 

25 

19 

17 

10 

28 

22 

31 

34 

28 

24 

24 

Denominate  Numbers  
Involution  and  Evolution. 

11 

20 

23 

34 

30 
1 

31 
i 

A 

29 

7 

22 
16 

367 


368  HOW  TO  TEACH  AEITHMETIC 

GRADES — Continued 
Subject  I       II     III     IV      V      VI   VII  VIII 

Decimal  Fractions 5       23       12         8         1 

Mensuration  .  6         8       10  '     11       12       14       14       12 


Multiplication  Tables  2         8       20       18 

7 

5 

7 

6 

Commission  and  Brokerage   

10 

11 

6 

Insurance   

10 

9 

6 

Percentage   <.  

7 

16 

13 

8 

Eatio  and  Proportion  1 

3 

8 

6 

9 

Partnership  

2 

7 

4 

Partial  Payments  

3 

5 

G.  C.  D.  and  L.  C.  M  

4 

6 

Longitude  and  Time  

4 

7 

1 

Profit  and  Loss  

7 

17 

2 

Taxes    

2 

14 

3 

Duties   

1 

13 

1 

Banking  

9 

6 

Exchange    

4 

4 

Simple  Interest  1 

2 

12 

23 

4 

Stocks  and  Bonds  

6 

8 

Business  Forms  

1 

4 

15 

6 

Simple  Accounts  .  .         3 

6 

5 

3 

3 

SOME  VALUABLE  BOOKS  FOE  SUPEEVISOES  AND 
TEACHEES   OF   AEITHMETIC 

"History  of  Mathematics, "  Ball;  The  Macmillan  Company,  Chi- 
cago. 

"History  of  Elementary  Mathematics/'  Cajorie;   The  Macmillan 
Company,  Chicago. 

"A  Brief  History  of  Mathematics, "  Fink;  The  Open  Court  Pub- 
lishing Company,  Chicago. 

"The  Educational  Significance  of  Sixteenth  Century  Arithmetic/' 
Jackson;   Teachers'  College  Bureau  of  Publications,  New  York. 

"Eara  Arithmetica/ '  Smith;  Ginn  &  Company,  New  York. 

"The   Hindu- Arabic   Numerals/'   Smith   and   Karpinski;    Ginn   & 
Company,  New  York. 

"The  Psychology  of  Number,"  McClellan  &  Dewey;  D.  Appleton 
&  Company,  Chicago. 


COURSES  OF  STUDY  359 

"A  Fundamental  Study  in  the  Pedagogy  of  Arithmetic/'  Howell; 
The  Macmillan  Company,  Chicago. 

"The  Number  Concept/'  Conant;  The  Macmillan  Company,  Chi- 
cago. 

"The  Teaching  of  Elementary  Mathematics/'  Smith;  Ginn  & 
Company,  New  York. 

"The  Teaching  of  Mathematics/'  Young;  Longmans,  Green  & 
Company,  New  York. 

"Iho  Teaching  of  Arithmetic,"  Smith;  Ginn  &  Company,  New 
York. 

"Methods  in  Arithmetic/'  Walsh;  D.  C.  Heath  &  Company,  New 
York. 

"The  Teaching  of  Arithmetic,"  Stamper;  American  Book  Com- 
pany, Cincinnati. 

"Special  Method  in  Arithmetic,"  McMurry;  The  Macmillan  Com- 
pany, Chicago. 

"Primary  Arithmetic,"  Suzzallo;  Houghton,  Mifflin  Company, 
Boston. 

"Arithmetical  Abilities  and  Some  Factors  which  Determine  Them," 
Stone;  Teachers'  College  Bureau  of  Publication^  New  York. 

"Standard  Tests  in  Arithmetic,"  Courtis.  [A  series  of  tests  pre- 
pared by  and  for  sale  by  S.  A.  Courtis,  Detroit,  Mich.] 

"Practice  in  the  Case  of  School  Children,"  Kirby;  Teachers'  Col- 
lege Bureau  of  Publication,  New  York. 

"Arithmetic  Supervision,"  Jessup  &  Coffman;  The  Macmillan 
Company,  (  hieago. 

"Number  Rhymes  and  Number  Games,"  Smith;  Teachers'  Col- 
lege Bureau  of  Publication,  New  York. 

Report  of  the  American  Commissioners  on  "The  Teaching  of 
Mathematics  in  the  Elementary  Schools,"  Bureau  of  Education, 
Washington,  D.  C. 

"How  to  Study  and  What  to  Study,"  Sandwick;  D.  C.  Heath  & 
Company,  Boston. 

RECREATIONS 

"Mathematical  Recreations,"  Ball ;  The  Macmillan  Company,  Chi- 
cago. 

"A  Scrap  Book  of  Mathematics,"  White;  The  Open  Court  Pub- 
lishing Company,  Chicago. 

"Mathematical  Wrinkles,"  Jones;  S.  I.  Jones,  Gunter,  Texas. 

"The  Canterbury  Puzzles,"  Dudeney;  E.  P.  Dutton  &  Company, 
New  York. 


INDEX 


Abacus,  5,  12 

Ability,  arithmetical,  21-31 

Abridged  Computations,  207-208 

Abstract  problems,  73-75 

Accuracy,  43-55,  102-104,  157-158 

Addition,  23-24,  29-30,  37-39,  47,  58- 
60,  150-159 

Ahmes,  5,  186 

Alcuin,  6 

Algebra,  8,  292,  345-348 

Aliquot  parts,  210,  217,  249,  331-332 

Allen,  148 

Alligation,  8 

Analyses,   48-51,   76-77,   82-91,   358- 
359 

Annual  interest,  118,  256-258,  261 

Applications,  121-122 

Approximations,  63-64,  207-208 

Arabic  numerals,  5 

Aristotle,  4 

Arithmetic, 
growth  of,  2-3 
among  Greeks,  5-6 
among  Romans,  6 
in  Hanseatic  League,  6-7 
during  Renaissance,  7 
since  Renaissance,  7-8 
racial  development  of,  8-9 
reasoning  in,  40-41 
scientific  studies  in,  17-41 
time  devoted  to,  120-121 
time  to  introduce,  134-136 
why  taught,  3,  112-113 
present  tendencies  in,  347-366 

Arithmetica,  5-6 

Assignments,  123-124 

Auerbach,  323 

Austrian  method,  8,  160 

Avoirdupois,  172-175 

Babylonians,  5,  10,  186,  240 
Banking,  271-277 
Bank  discount,  274-276 
Bibliography,  368-369 
Binary  system,  9 
Bobbitt,  39 


Boethius,  6 

Bonds,  279-281 

Bonser,  40-41 

Bookkeeping,  277 

Brooks,  195,  198,  210 

Brown,  105 

Business  practice,  271-281 

Cancellation,  120 

Casting  out  nines,  59-63 

Chocks,  56-65 

Circulating  decimals,  117,  210 

Commercial  discount,  228-236 

Commission,  236-239 

Compound  interest,  257-261 

Computation,  119-120,  356-357 

Conan.t,  9 

Concentric  circle  plan,  133,  364 

Concrete  problems,  73-75,  169-170 

Copying  figures,  54-55 

Corporations,  277-280 

Correlation,  28-31,  136-137 

Corson,  169 

Counting,  8-9,  10,  144-145,  149-150 

Course  of  study,   34,   132-133,   366- 

368 

Courtis,  34-40 
Cube  root,  118,  290 
Culture  value,  112-113 

Date  line,  342-344 

Decimal  fractions,  8,   11,   117,   195- 

212 

Definitions,  184-185,  361 
De  Morgan,  84,  196 
Denominate  numbers,  117,  171-181 
Dewey,  210,  213,  215,  364 
Discipline,  formal,  24-26,  131-134 
Discount,  bank,  274-276 
Discount,  commercial,  228-236 
Discount,  true,  117 
Division,  29-30,  37-39,  47-48,  62-63, 

167-169 

Drill,  92-109,  128 
Duties,  270 


371 


372 


INDEX 


Earhart,  126 

Easier  problem,  6 

Egyptians,  5,  10 

Elimination,   115-120,   180,   185-186, 

210,  276,  290,  354-356 
Equation,  299-300,  345-348 
Equation  of  payments,  7,  117 
Estimating  results,  63-64 
European  schools,  208,  315,  320 
Evolution,  290-298 
Examinations,  127-130 
Examples,  77-78,  357 
Explanations,  124-125 

Factoring,  291 

figures,  45-46,  54-55 

Formal  discipline,  24-26,  131-134 

Formula  method,  249 

Fractions,    common,    10,    11,    48-50, 

117,  182-194,  208-211,  214,  217 
Fractions,  decimal,  see  Decimals 
Functions,  320 
Fundamental  operations,  148-170 

Gallon,  174 

Games,  140-142,  166-167 

Geometry,  314-318 

Germany,  315 

Grading  papers,  66-70 

Graphs,  319-323 

Greatest  common  divisor,  117 

Greeks,  5-6,  10,  187,  195,  241 

Grube,  14-15,  363-364 

Hanseatic  League,  6-7 
Hanus,  355 

Hindu  numerals,  5,  7,  11 
Hindus,  5,  187 
History  of  arithmetic,  1-16 
•History   of   common    fractions,    186- 

188 
Humanists,  4 

Insurance,  261-266 
Interest,  awakening,  349,  352-354 
Interest,  simple,  239-255,  258,  261 
Interest,  annual,  118,  256-257,  258, 

261 

Interest,  compound,  257-261 
Investigations,  17-41,  104-109,  362- 

363 


Involution,  290 
Isidore,  6 

Jackson,  7 

Kepler,  197 
Kirby,  109 
Kranckes,  13-14 

Laisant,  113 

Least  common  multiple,  117 

Leibnitz,  9 

Liliwati,  187,  188 

Literal  arithmetic,  312,  345-348 

Logistica,  5-6 

Longitude  and  time,  51,  333-344 

McLellan  and  Dewey,  210,  213    215 
364 

McMurry,  126,  360 

Marking  papers,  66-70 

Marking  goods,  234-236 

Mensuration,  47-48,  51,  167,  301-318 

Mental    arithmetic,    see   Oral   arith- 
metic 

Mental  discipline,  24-26,  113,  131 

Methods,  11-16 

Methods  of  study,  125-126 

Metric  system,  282-289 

Mixed  numbers,  47 

Models,  310 

Multiplication,    29-30,    37-39,    47-48 

61-62,  163-167,  326-332 
Multiplication  tables,  163-166 
Myers,  44,  351 

Napier,  8,  195 

Negotiable  paper,  272-276 

Nines,  casting  out,  59-63 

Notation,  8-9,  146 

Number  games,  140-142,  166-167 

Numbers,  45-46 

Number  systems,  9 

Object  teaching,  12-15,  136-140,  142- 

145,  149-150,  185,  188-189 
Obsolete,  115-119,  276,  354-356 
One-to-one  correspondence,  8 
Oral  arithmetic,  75-76 

Papers,  marking,  66-70 
Partial  payments,  276 
Partition,  167 


INDEX 


373 


Pellos,  197 

Percent,  50-51 

Percentage,  117,  213-222 

Pestalozzi,  5,  7,  12-13 

Plato,  4,  5 

Play,  see  Games 

Primary    arithmetic,    131-147,    148- 

170 
Problems,  71-81 

analysis    and    solution    of,    48-51, 
100-101,  118-119,  357-359 

applied,  169-170 

checking,  64-65 

material,  350-354 
Profit  and  loss,  223-228 
Progression,  118 
Promissory  notes,  272-27G 
Proportion,  299-300 
Psychology,  128 
Pythagoras,  4 

Quadrivium,  6 

Rates  of  interest,  241-246 

Rationalizing,  359 

Ratio,  299-300 

Reasoning,  29-30,  40-41,  69-70 

Rechenmeister,  6 

Renaissance,  7-8 

Revenue,  270 

Reviews,  127-129 

Rhyming  arithmetics,  82-84 

Rice,  17-20,  34 

Romans,  6,  10,  142,  18V ,  195,  241 

Roman  numerals,  7,  146-147 

Roots, 

square,  290-298 

cube,  118,  290 
Rules,  82-91,  213,  215,  223 
Rule  of  three,  8,  300 

Scales  of  counting,  9 

Scientific  investigations,  17-41,  104- 

109 

Separatrix,  198 
Series,  8 

Short  cuts,  324-332 
Similar  figures,  316-318 
Simple  interest,  239-257 
Six  per  cent  method,  247-248 
Smith,  5,  7,  44,  215,  347 
Solon,  4 


Sophists,  5 

Speed,  53-54,  76,  102-104,  157-158 

Speer,  15-16 

Spencer,  82,  84,  87-88,  360 

Spiral  method,  133,  364 

Square  root,  290-298 

Standards  in  arithmetic,  34-40 

Standard  time,  338-342 

Starch,  31 

Stocks  and  bonds,  277-281 

Stone,  21-34,  54,  68 

Study,  125-126 

Study,  course  of,  366-368 

Subtraction,   29-30,   37-39,   60,    160- 

163 

Suzzallo,  138,  355 
Symbols,  9-10,  45-46,  145-147,  148, 

197 

Tables, 

addition,  151-153 

denominate  numbers,  173-176,  180 

interest,  252,  259,  275 

mortality,  264 

multiplication,  162-166 

subtraction,  160-163 
Taxes,  266-270 
Taylor,  36 
Teachers,  56-57 
Tendencies,  345-348,  349-366 
Text-books,  7,  71-72,  363-366 
Thorndike,  28,  29,  104 
Tillich,  13 
Time,    32-34,    51,    120-121,    134-136, 

333-344 

Troy  weight,  172-175,  180 
True  discount,  117 

Unitary  analysis,  90-91 
Unity,  122,  359-361 
Usury,  240-243 

Von  Busse,  12 

Waste  in  arithmetic,  110-130 
White,  260 
Wilson,  367 

Yerkes,  147 
Young,  85,  347 
Zero,  10 


SCHOOL  MANAGEMENT 

By  ALBERT  SALISBURY,  Ph.  D. 

President  of  the  Whitewater  State  Normal  School,  author  of 
"The  Theory  of  Teaching,"  etc. 

Cloth,  12mo.,  196  pages,  $1.00 

This  book  represents  the  fruits  of  a  lifetime  spent  in  the  schools 
and  in  the  training  of  teachers.  School  conditions  have  changed 
greatly  in  recent  years,  and  books  on  school  economy  which  were 
excellent  a  few  years  ago  are  now  antiquated.  Much  more  is 
demanded  of  the  teacher  than  formerly.  He  has,  in  fact,  become 
an  official  of  the  state,  with  larger  functions  and  a  greater  need 
for  intelligence  concerning  those  functions  than  the  old-time 
pedagog. 

While  endeavoring  to  recognize  this  newer  conception  of  the 
teacher's  office,  and  the  greater  burden  which  it  imposes,  it  has 
been  the  desire  of  the  author  to  make  a  small  book  rather  than  a 
bulky  one,  excluding  padding  and  time-honored  common-place. 
The  book  is  intended  to  serve  the  needs  of  young  teachers  and 
those  in  preparation  for  the  work,  and  clearness  has  been  aimed 
at  rather  than  profundity. 

Testimonials 

Frank  A.  Weld,  Pres.  State  Normal  School,  Moorhead,  Minnesota 

"I  have  been  reading  Salisbury's  'School  Management*  with  great  interest. 
It  is  a  stimulating  book  and  should  find  its  way  into  many  Normal  schools." 

Dr.  A.  E.  Winship,  Boston,  Mass. 

"I  have  spent  more  time  on  'School  Management*  than  I  intended,  because 
I  have  enjoyed  it  more  than  I  expected  to.  It  is  in  the  fullest  sense  a  notable 
book.  It  gives  what  is  needed  in  the  least  space,  in  the  best  spirit,  and  in  a 
most  enjoyable  style.  I  am  charmed  with  it." 


the  vigorous  treatment  and  language  or  LJr.  Salisbury  s  School  Management. 
The  book  includes  all  the  necessary  new  features,  and  omits  quite  as  wisely  as 
it  includes.  It  is  right  in  size,  covers  the  necessary  ground,  and  occupies  safe 
and  sane  positions." 

Wisconsin  Journal  of  Education,  Madison,  Wisconsin. 


er  or  writing  maK.es   ocnooi  management   a  volume  luu  01  iiiccii 
aluable  addition  to  the  present  literature  on  this  important  subject, 


Sent  postpaid  to  any  address  on  receipt  of  price. 
Liberal  discount  to  classes. 

ROW,  PETERSON  &  CO.,  Publishers 

CHICAGO,  ILLINOIS 


THIS  BOOK  IS  DUE  ON  THE  LAST  PATE 
STAMPED  BELOW 


LrD  21-100m-7,'33 


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